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“Beginning Jan. 20, the administration will give the world’s leading state sponsor of terrorism billions of dollars while allowing the mullahs to keep their illicit nuclear infrastructure in place,” Mr. Kirk said in a statement. “I am worried the administration’s policies will either lead to Iranian nuclear weapons or Israeli airstrikes.”

The White House has vowed to veto any legislation calling for more sanctions against Iran, arguing that such a move would threaten the entire agreement. Indeed, Iranian Deputy Foreign Minister Abbas Araghchi was quoted by state media Sunday saying any new sanctions would void the deal.

“Imposing additional sanctions now will only risk derailing our efforts to resolve this issue peacefully, and I will veto any legislation enacting new sanctions during the negotiation,” Mr. Obama said, reiterating his long-standing position.

Mr. Dubowitz, however, argues that further sanctions are needed to keep the process moving forward. He points to research showing that Iran’s economy has begun to recover as international investors see the nation as a safe place to do business, with the grip of American economic sanctions easing.

If that trend continues, Iran could change course, ditch the deal and go back to pursuing a nuclear weapon at all costs, Mr. Dubowitz said.

“Congress rightly perceives that it is only sanctions pressure that has persuaded Iran to come to the negotiating table and it is only the threat of new sanctions that will persuade Iran to conclude a final deal,” he said.

Guy Taylor contributed to this report.