WASHINGTON—The White House and Senate negotiators sought to keep a bipartisan infrastructure deal on track Sunday after its future was cast into doubt when President Biden made and then withdrew a suggestion that he would veto the proposal if lawmakers failed to pass a separate antipoverty package favored by Democrats.

Several Senate Republicans who reached agreement with Democratic lawmakers and the president on infrastructure appeared mollified Sunday by Mr. Biden’s statement a day earlier walking back his comments tying the fate of the roughly $1 trillion infrastructure agreement to the larger piece of legislation that could cost trillions more.

“I am glad they have now been delinked and it’s very clear that we can move forward with a bipartisan bill,” said Sen. Rob Portman (R., Ohio), on ABC’s “This Week.” He added: “We were glad to see them disconnected.”

Said Sen. Mitt Romney (R., Utah) on CNN’s “State of the Union”: “I do trust the president.”

A group of 21 senators, including 11 Republicans and 10 members of the Democratic caucus, had lent their support to the deal, which provides $579 billion above expected federal infrastructure spending. The bill includes funding for improvements to roads, bridges, transit, airports and enhanced infrastructure for broadband, water and electric vehicles.

The White House sought to reassure lawmakers on the future of the infrastructure bill, a central plank of the Biden agenda that already faced a complex path to passage in the narrowly divided Congress.

In the Senate, Democrats hope to pass the bipartisan bill by drawing enough Republican votes to reach the 60-vote threshold required for most legislation to advance. They also hope to pass the broader antipoverty bill under a special process tied to the budget, known as reconciliation, that requires only a simple majority.

Cedric Richmond, a former Louisiana congressman who serves as a top aide to Mr. Biden, said on “Fox News Sunday” that the president was “making sure that we’re talking about the issue and not the process. The issue is we have to invest in our country with crumbling bridges and roads, and we’re going to deal with it.”

Asked on CNN whether the president would sign the bill on its own without the budget reconciliation package, Mr. Richmond wouldn’t say. “I don’t think it’s a yes or no question,” he said.

Sen. John Barrasso (R., Wyo.), the chairman of the Senate Republican Conference, said Republicans would need more reassurance from the president that the bill wouldn’t be handcuffed to the larger antipoverty package.

“Look, I’m a doctor, and I will tell you, you can get whiplash by trying to follow Joe Biden on this,” Mr. Barrasso said on Fox News’s “Sunday Morning Futures.”

President Biden’s infrastructure plan calls for nontraditional projects like the removal of some highways. What Democrats want for cities like Baltimore says a lot about the president’s goals in the next wave of development. Photo: Carlos Waters/WSJ The Wall Street Journal Interactive Edition

As Mr. Biden seeks to win passage of the infrastructure bill, he is trying to balance his interest in gaining a bipartisan achievement with Republicans with demands from liberal members of his party, who have pushed for a $6 trillion antipoverty proposal, the American Families Plan, that includes an array of social-welfare items.

Senate Majority Leader Chuck Schumer (D., N.Y.) plans to bring up in July both the bipartisan bill and the budget resolution, which will set the overall spending levels and parameters of the larger package. Lawmakers expect to work on passing that proposal in September, following Congress’s August break. House Speaker Nancy Pelosi (D., Calif.) said last week she wouldn’t bring up the bipartisan infrastructure bill unless the antipoverty package had passed the Senate.

A 50-50 divide in the Senate, with Vice President Kamala Harris casting any tiebreaking vote, and a narrow majority in the House for Democrats have complicated those efforts.

Rep. Alexandria Ocasio-Cortez (D., N.Y.) said on NBC’s “Meet the Press” that House Democrats are committed to making sure that Democrats pass the broader antipoverty plan, which expands child care, Medicare, affordable housing and other programs. Democrats hope to pay for the continuing programs by boosting taxes on businesses and upper-income households.

“It’s very important for the president to know that House progressives and I believe the Democratic caucus is here to make sure he doesn’t fail,” she said. The president shouldn’t be limited by Republicans, she said, in light of the Democratic majority.

Sen. Joe Manchin (D., W.Va.), a key swing vote in the Senate, on Sunday signaled he might be willing to support a broader spending package of $2 trillion beyond the infrastructure proposal. Mr. Manchin said on ABC that he would support a reconciliation bill that included raising the corporate tax rate to 25% and taxes on capital gains to 28%.

“I want to make sure we pay for it. I do not want to add more debt on. So if that’s $1 trillion, or $1.5, or $2 trillion, wherever that comes out to be over a 10-year period, that’s what I would be voting for,” he said.

—Stephanie Armour and Rachael Levy contributed to this article.

Write to Ken Thomas at ken.thomas@wsj.com