President Donald Trump on Tuesday said the United States could soon reach an agreement with France that would preserve the 2015 Iran nuclear deal.

Trump made his comments during a news conference with French President Emmanuel Macron, who is trying to dissuade his American counterpart from pulling out of the accord. The push appeared to bear fruit on Tuesday as the leaders suggested a deal could be announced ahead of a crucial deadline in just under three weeks.

"We understand each other, and we'll see how that comes out," Trump told reporters on Tuesday. "We could have at least an agreement among ourselves fairly quickly. I think we're fairly close to understanding each other, and I think our meeting, our one-on-one, went very, very well."

Macron on Tuesday outlined a three-point plan that addresses several — but not all — of Trump's complaints about the nuclear accord. The plan, which is short on details, presumably forms the basis for ongoing negotiations between the United States and Europe.

The deal in question is the Joint Comprehensive Plan of Action, an agreement reached between Iran and six world powers in 2015 that placed limits on Iran's nuclear program in exchange for sanctions relief.

Trump has called the accord the worst deal ever. Earlier this year, he said he would withdraw the United States from the pact and restore sanctions on Iran if he could not reach a deal to toughen the terms of the agreement with France, Germany and the U.K., all of which support the accord.

By May 12, Trump must decide whether to waive or restore sanctions on Iran. On Tuesday, he suggested Macron knows where he stands.

"Nobody knows what I'm going to do on the 12th," Trump said before turning to Macron and saying, "Although Mr. President, you have a pretty good idea, but we'll see."

The Trump administration and congressional Republicans are seeking several changes to the deal, which was negotiated by the Obama administration. They want to prohibit Iran from testing ballistic missiles, expand international inspectors' access to Iranian military facilities and prevent key aspects of the deal from expiring.

Trump also wants to tie the fate of the nuclear deal to a plan to address Iran's role in several conflicts in the Middle East, including in Iraq, Syria and Yemen.

On Tuesday, Macron said France now wants to build on the progress of the 2015 nuclear deal, which he called the first of four pillars to stabilize the Middle East. The plan he presented overlaps with Trump's priorities.

Moving forward, Macron wants the international community to begin addressing potential restrictions on Iran's nuclear program beyond 2025, when key aspects of the deal expire.

However, he was careful not to commit to making these so-called sunset clauses permanent — something Trump wants and Iran fiercely opposes. The fate of the sunset clauses has been a major sticking point in negotiations between the United States and Europe.

Macron also said future negotiations must tackle Iran's ballistic missile program and its support for the Syrian regime, Houthi rebels in Yemen and the Lebanese political and militant group Hezbollah.

James Rubin, former U.S. assistant secretary of State for public affairs, said the outcome of the meeting is vague, but it appears Macron is trying to overcome a diplomatic spat by enlarging the issue.

"What Macron has done is essentially try to hide the agreement among a lot of diplomatic terminology that suggests there's going to be a larger agreement that has four pieces to it," he told CNBC. "There's one old agreement that's going to be down at the bottom, and then we're going to build all this new great stuff on top of it."

However, he said none of that stuff is likely to get built soon because Iran opposes altering the deal.

Richard Nephew, the lead sanctions expert for the U.S. team that negotiated the nuclear deal, cautioned against drawing conclusions, noting that Trump has flip-flopped after appearing to announce a policy decision in the past.

"It sounds optimistic, but this is not the first time we've seen real progress alleged and then we've backslid," said Nephew, now a senior research scholar at Columbia University's Center on Global Energy Policy.

"This is one discussion with one person who wants to keep the JCPOA. Let's fast-forward 20 minutes from now to a meeting with" secretary of State nominee Mike Pompeo and national security advisor John Bolton, he said, referring to two of the administration's foreign policy hawks.

Still, critics of the deal were quick to warn Trump against losing his resolve.

"While seeking to fix the JCPOA nuclear deal, President Trump should beware of a 'fake fix' from Europe on Iran's ballistic missiles," said Behnam Ben Taleblu, Iran research fellow at the Foundation for Defense of Democracies, a think tank that opposes the current deal and has advised the Trump administration on renegotiating it.

Taleblu notes that the Europeans have signaled they would support sanctions that target only long-range ballistic missiles. He called that an example of "a Band-Aid solution" that "would contain future promises to prevent an Iranian ICBM rather than restricting the bulk of their nuclear-capable arsenal today."

The market also appeared to take Trump's remarks as a softening of the U.S. position on Iran. Oil prices turned lower after his comments, shedding some of the recent gains that were fueled by concerns that renewed sanctions would disrupt crude supplies from Iran, OPEC's third-biggest producer.

John Kilduff, founding partner at energy hedge fund Again Capital, said Trump's tone came as a surprise to the market.

"We were more expectant of the harsh rhetoric he started with than that he would intimate there's a deal," he said.

— CNBC's Patti Domm contributed to this story.

Trump: Agreement with France on Iran nuclear deal could come 'fairly quickly'