VIENNA—Tensions in the nuclear talks between Iran and six powers have boiled over in recent days, producing heated exchanges among foreign ministers as Washington and Tehran struggled to overcome remaining hurdles to a final agreement, according to people involved in the talks.

The German and British foreign ministers returned to the Austrian capital Wednesday evening as Western diplomats insisted a deal was still possible in coming days. However, time was running out for the agreement to be sealed before a deadline this week, which would give the U.S. Congress an extra month to review a deadline.

People close to the talks have warned that the longer Congress and opponents of the diplomacy get to pick over an agreement and galvanize opposition, the greater the political risks for supporters of the process, which aims to block Iran’s path to nuclear weapons in exchange for lifting tight international sanctions.

U.S. officials have insisted this week they don’t feel under pressure to get a deal by the congressional deadline, which arrives at midnight Thursday (6 a.m. Friday in Vienna.)

Russian Foreign Minister Sergei Lavrov said Thursday that he may rejoin the talks on Friday, the clearest public indication yet that the negotiations would run beyond the congressional deadline. Speaking on the sidelines of a summit of Brazil, Russia, India, China and South Africa, he also said there are no “insuperable problems” in the nuclear talks.

Still, over the past day, Western officials and Iranian media have outlined tense exchanges between the negotiating teams that took place Monday evening, at a point where the talks appeared close to stalling. At the time, negotiators were working toward a Tuesday deadline for a deal.

Iranian media cited a shouting match between Secretary of State John Kerry and Iran’s Foreign Minister Javad Zarif, which forced a U.S. aide to intervene and warn the ministers their voices could be heard, beyond the meeting room, according to the account.

Two Western officials and Iranian media also said that during discussions among foreign ministers on Monday, European Union foreign policy chief Federica Mogherini threatened to leave the negotiations, prompting a shouted response from Mr. Zarif to never threaten an Iranian. Russian Foreign Minister Sergei Lavrov added “nor a Russian,” according to the accounts.

It was “one of these frank exchanges that can happen when one negotiates difficult issues,” said one Western official, who added that the tension was defused quickly.

“There was no slamming of doors but it was a very heated exchange of views” on Monday evening, said a second senior Western diplomat.

Spokesmen for the U.S. State Department didn’t respond to requests for comment.

Iranian Foreign Ministry spokeswoman Marzieh Afkham also played down the incident, which was widely trailed in Iranian media.

“As you know, there are moments in any negotiations. Dr. Zarif has a very good working relations with all his seven counterparts,” she said in an emailed comment. In a Twitter message on Wednesday, Mr. Zarif said: “With mutual respect, anything possible.”

Western officials said that Monday evening’s meeting between the foreign ministers was a crucial point in the talks, a moment where it seemed that an agreement would be impossible without a major shift by one or both sides. It also came as temperatures climbed to 100 degrees in the Austrian capital.

That night, the foreign ministers of the U.S., U.K., France, Germany, Russia, China and Iran held two meetings, each between one and two hours long, and the second lasting into the early morning hours of Tuesday.

Later on Tuesday, President Barack Obama told Senate Democrats in Washington that he now believed there was less than a 50-50 chance of a final deal with Iran, according to a person familiar with discussions.

Ms. Mogherini effectively stopped the clock on the talks by announcing Tuesday that Iran and the six powers had agreed to keep on negotiating, without formally extending the deadline. U.S. officials said then that the negotiations could still go either way, but they had no immediate plans for Mr. Kerry and the U.S. team to return home.

The White House, meanwhile, was laying the groundwork for an alternative to either a breakthrough or a breakdown by suggesting talks could continue beyond Tuesday’s deadline while keeping in place a 2013 interim nuclear agreement that provided Iran with limited sanctions relief in exchange for rolling back parts of its program.

With foreign ministers out of town, Wednesday was a chance for officials to continue technical work and prepare a final push to reach an agreement. French Foreign Minister Laurent Fabius arrived back in Vienna on Thursday morning.

People close to the talks say discussions have advanced since Monday, with the two sides exploring a set of possible trade-offs on some of the final big issues.

Those include the fate of a United Nations arms and ballistic missile ban on Iran, how and when Tehran will address its past nuclear work, the timing of sanctions relief and assurances that United Nations inspectors will have access to any sites of concern in Iran under a deal.