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 reassured 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.”

The negotiations at the start of a summerlong legislative blitz represent the biggest test yet of Mr. Biden’s ability to achieve a bipartisan victory in the first year of his presidency. The former vice president has pointed to his three decades in the Senate as proof of his deal-making abilities at a time of deep partisan polarization, increasing the stakes for Mr. Biden, should the deal collapse.

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 and airports as well as enhanced infrastructure for broadband, water and electric vehicles.

The infrastructure bill, a central plank of the Biden agenda, 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 bill, which expands child care, Medicare, affordable housing and other programs 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 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.”

While Mr. Biden’s earlier linking of the two initiatives drew a rebuke late last week from Senate Minority Leader Mitch McConnell (R., Ky.), Sen. Bill Cassidy (R., La.) said he thought Mr. McConnell would support the plan “if it continues to come together as it is.”

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