A person familiar with the meeting confirmed that President Biden met with Senate Majority Leader Chuck Schumer and Senator Joe Manchin in Delaware on Sunday to try to reach an agreement on the president’s social spending plan.

According to a White House official, Mr. Biden hosted the two for breakfast at his home. When asked about the meeting with Manchin on Monday, Trump told reporters that there are “some things to work out but it went well,” and that he hopes to have a deal on the sweeping package before leaving for Europe on Thursday for summits in Rome and Glasgow, Scotland.

“It would be very, very positive if we could get it done before the trip,” Mr. Biden said before leaving Delaware for New Jersey, where he will pitch his domestic policy agenda and campaign with Democratic Governor Phil Murphy, who is up for reelection in November.

Democrats are attempting to reach an agreement on a framework for Mr. Biden’s domestic policy agenda, which will necessitate support from Manchin and Arizona Senator Kyrsten Sinema, both moderate senators who objected to the package’s initial $3.5 trillion price tag.

However, there is disagreement between Manchin and Democratic leaders on the topline figure, as the West Virginia senator has said he supports a $1.5 trillion plan, while Mr. Biden has set a price range for the legislation at around $2 trillion. Along with trimming the social spending package, Democrats are working out the details of how to pay for it, as the White House faces opposition from Sinema over its plan to raise the corporate tax rate and the top income tax rate for the wealthiest Americans.

Mr. Biden’s expansive domestic policy package was supposed to include plans for paid family leave, free community college, universal pre-K, and expanding Medicare to include dental, vision, and hearing care. However, during a town hall meeting on Thursday, President Obama stated that Manchin is opposed to Medicare expansion and free community college. Mr. Biden also stated that his original plan for 12 weeks of paid family leave had been reduced to four weeks.

The president, on the other hand, expressed optimism that Democrats in both chambers could reach an agreement on a framework for the package that would garner Manchin and Sinema’s support.

“What we’re trying to do is get to the point where I can present to the Senate — and they can vote on it — and the House a serious, serious piece of legislation that changes the dynamic for working-class and middle-class people in America, and begins to have the very wealthy and corporations just begin to pay their fair share,” he said Thursday.

Democratic leaders hope to reach an agreement on the social spending plan before Mr. Biden travels to Europe, where he will attend the Group of 20 Leaders’ Summit in Rome and the United Nations Climate Summit in Glasgow. A deal on the package’s details would pave the way for the House to vote on a $1 trillion bipartisan infrastructure bill, which has already passed the Senate, before the president leaves Washington, giving him a key legislative victory before meeting with world leaders.

House Speaker Nancy Pelosi said on Sunday’s “State of the Union” that Democrats have “agreed to and written” the majority of the social safety net bill. The “plan,” she said, is for the House to vote on the physical infrastructure bill next week and Democrats to agree on a framework for the broader package.