WASHINGTON — President Obama, in advance of his victory in Congress on Thursday on the Iran nuclear deal, focused on centrifuge numbers, uranium stockpiles and the breakout time to build a bomb. But now, as the agreement is carried out, he will face a new battle over how stringently to impose economic sanctions on Iran.

To reward Iran for imposing constraints on its nuclear program, the United States agreed to lift many of the crippling sanctions that have blocked the country’s integration into the world economy. But to win over wary Democrats, Mr. Obama promised that he would maintain — and perhaps even increase — sanctions to punish Iran for terrorism, human rights abuses and other “destabilizing activities in the region.”

Many lawmakers have indicated they would like to go further, and they are considering legislative proposals that include renewing the current sanctions against foreign companies that invest in Iran’s energy industry. Mr. Obama would waive them as long as Iran complied with the nuclear accord, but these sets of actions would be a signal that Iran is not to be trusted and that sanctions could be restored rapidly.

Another proposal considered by lawmakers would seek to discourage Western companies from doing business with any Iranian firm in which the Islamic Revolutionary Guards Corps has even a minority stake; that would be done by officially designating the group a foreign terrorist organization.

The first sanctions are to be lifted after Iran reduces the size of its stockpile of low-enriched uranium and meets other requirements, a process many American experts say could take at least six months — although the Iranians insist they can do it much sooner.

“There is a tension between ensuring that Iran has an economic incentive to keep the agreement with its nuclear constraints and ensuring companies keep doing business with Iran, while at the same time policing Iran’s support for terrorism with sanctions,” said Richard Nephew, who worked on Iran sanctions issues for the Obama administration and is now a fellow at the Center on Global Energy Policy at Columbia University.

At the Treasury Department, much of the promised sanctions relief would be facilitated by withdrawing the threat that foreign companies and banks doing business with Iran would be subject to a range of punishments that would restrict their business in the United States — what the department calls “secondary sanctions.”

When it comes to American sanctions relief under the accord, “it almost entirely concerns transactions between non-U.S. companies, non-U.S. banks and Iran,” Adam J. Szubin, the acting Treasury under secretary for terrorism and financial crimes, said in an interview. “It is the U.S. sanctions targeting the conduct of foreign companies that are due to be relieved under the deal.”

So-called U-turn transactions — that is, transactions between foreign companies in which funds are routed through a United States bank to convert them to dollars — also would not be allowed. Such transactions are a fundamental part of international commerce, are carried out millions of times a day and are central to full participation in a world trading system still dominated by the dollar. The Obama administration has also emphasized that there are provisions to “snap back” a range of American and international sanctions in the event of Iranian cheating.

Even after the nuclear accord takes hold, Iran will be blocked from business with the United States because of a longstanding American trade embargo, as well as terrorism and human rights sanctions — all of which will remain in force. The exceptions include the export of passenger planes and aircraft parts to Iran, and the import of Iranian carpets, pistachios and caviar.

The talk of imposing new sanctions has not gone unnoticed in Tehran. In July, the Iranian government warned in a statement that if the West pushed too hard, then the deal, or least some elements of it, could be off.

Still, the agreement would give Iran access to the banking system outside the United States as well as to foreign companies and markets, both to give Iranians an incentive to honor the accord and, proponents of the deal hope, to strengthen the hand of President Hassan Rouhani and other moderates who have promised economic and political change.

For example, of the nearly 690 Iranian banks and companies the Treasury Department put on its sanctions list because of links to Iran’s nuclear or missile programs, nearly two-thirds will be removed on “Implementation Day,” when the agreement goes into effect. An additional 43 Iranian individuals and entities would be removed eight years after the accord takes effect.

The agreement will allow foreign subsidiaries of American companies to do business with firms that are taken off the sanctions list. But they will require a license from the Treasury Department, which will define the scope of the permitted activity — and the American parent company cannot in any way support this activity.

“If you are really talking about a small U.S. subsidiary, that last point ends up being prohibitive,” Mr. Szubin, the Treasury official, said. “The companies that will be able to operate under the provision are likely to be the larger multinational firms that have large subsidiaries in Asia, Africa or the Middle East. That is, subsidiaries that can independently generate and support trade with Iran.”

Navigating a path through the remaining sanctions will not be easy, however, and some legal experts said erecting a firewall high enough to separate even a large subsidiary from its parent company may be challenging.

Given the hold the Revolutionary Guards have on crucial sectors of Iran’s economy, including construction, banking and energy, some experts say the sanctions relief may strengthen their hold — unless the Iranian economy is overhauled. Given the complexity of the sanctions, others wonder if Iran will realize all the economic benefits it is counting on. The Revolutionary Guards are likely to keep establishing front companies, prompting a cat-and-mouse game between Iran and the Treasury Department.

“Foreign companies conducting business in Iran after sanctions relief begins will need to carefully navigate those sanctions that endure, including the reality that dealings with Iranian actors that remain designated could still invoke U.S. secondary sanctions,” said Eytan J. Fisch, a former Treasury Department official who represented the agency in the Iran talks and is now at the law firm Skadden, Arps, Slate, Meagher & Flom.

Adding to the complexity, the United States and the European Union have maintained separate lists of sanctioned entities. Khatam al-Anbiya, the largest construction firm in Iran and a source of revenue for the Revolutionary Guards, would come off the European Union’s sanctions list after eight years. But it would remain on the American list indefinitely, and any European company that did business with it could be subject to a range of secondary sanctions.

Experts wonder how severely a future administration might punish a European company that did business with an Iranian firm that had been removed from the European Union’s list or carried out transactions with a subsidiary that was not specifically sanctioned by the Treasury Department.

“An additional challenge for companies everywhere in the world that have operations in both the United States and Europe will be the divergent sanctions regulations concerning Iran come Implementation Day,” said Adam M. Smith, a former Treasury and White House official who worked on sanctions and is now at the law firm Gibson, Dunn & Crutcher. “Firms will need to assess how to comply with two different — and potentially conflicting — sets of requirements.”

Even as the accord moves forward, Mark Dubowitz, an Iran sanctions expert at the Foundation for Defense of Democracies, which opposes the nuclear accord, said that lawmakers would most likely push for a tougher stance, including some legislators who support the agreement but want to hedge their bets. One idea is to designate the Revolutionary Guards as a terrorist organization and thus discourage business with the hundreds of companies in which the Guards are believed to have a minority stake or over which they have influence through board membership or through their role as key executives.

“It is not enough to merely target the Quds Force, which is controlled by the Revolutionary Guards, for its terrorist activities. The designations of the Guards would create a whole new dimension of reputational and legal risk,” he said. “It would result in a powerful deterrent for international banks and companies contemplating doing business with the most dangerous element of the regime.”

Some Obama administration officials view that effort as a thinly disguised move to throw a monkey wrench into the agreement and insist the Treasury Department is prepared to be vigilant about curbing Iran’s nonnuclear muscle flexing without the extra help from Congress.

Mr. Szubin said that the administration was determined to crack down on the Quds Force’s financial schemes and was already consulting on new financial targets.

“I just got back from Israel, where I talked with their security forces, their intelligence and their military about precisely this,” he said. “There is a whole lot we need to be doing jointly to target Hezbollah’s lines of support, the Quds Force’s lines of support — how they are procuring parts for U.A.V.s, how are they funneling money from their diaspora communities. We have the tools.”

This article was written by Michael R. Gordon for The New York Times on Sep. 11, 2015.