The bipartisan discussions have had a soft Memorial Day deadline, perhaps leading to another week of back-and-forth before Democrats start trying to assemble votes on their own.

A partisan strategy wouldn’t guarantee that the entire White House plan could become law; Democrats have slim majorities in the House and Senate, and they would need to figure out just how much taxing and spending those lawmakers can accept.

“We would like bipartisanship, but I don’t think we have a seriousness on the part of the Republican leadership to address the major crisis facing this country,” Sen. Bernie Sanders (I., Vt.), the chairman of the Senate Budget Committee, which would assemble any one-party infrastructure spending package, said on CBS’s “Face the Nation” on Sunday. “If they’re not coming forward, we’ve got to go forward alone.”

Sen. Susan Collins (R., Maine), a key Republican whose vote would likely be needed for any bipartisan deal, said the administration’s insistence on social spending makes an infrastructure bill difficult to reach.

“Negotiations should continue, but it’s important to note that there’s a fundamental difference here and at the heart of the negotiations is defining the scope of the bill: What is infrastructure?” she said on ABC’s “This Week” on Sunday. “I think we’re still pretty far apart.”

White House officials last week exchanged proposals with a group of Republican senators led by Sen. Shelley Moore Capito (R., W Va.). The latest White House offer reduced its proposed spending from $2.3 trillion to $1.7 trillion. That is still more than triple the GOP proposal, and the administration continued to stress the importance of including funding for elderly and disabled Americans.

“The human investment is important to the president,” Cedric Richmond, a White House aide, said on CNN’s “State of the Union” on Sunday. “He will not let inaction be the final answer, and he’s going to continue to invest in the American people and the infrastructure so that we can win tomorrow.”

Republicans say they would prefer to keep a bill smaller and more focused on items such as roads, bridges, ports and broadband.

The latest White House offer reduced the cost, but part of that came by moving some items to other legislation. And the administration is insisting on items such as renewable-energy tax credits, worker training, environmental cleanups and funding for veterans’ hospitals.

Mr. Richmond said President Biden is willing to let the bipartisan talks continue. Mr. Biden has been traveling the country to promote his plan. On Thursday, he will deliver remarks on the economy in Cleveland as part of those efforts.

Mr. Biden pitched his infrastructure plan during a tour of the Ford Motor Co. Rouge Electric Vehicle Center in Dearborn, Mich., last week. The plan includes funding for electric vehicles, and Mr. Biden has contended that his proposals will help create jobs and make the U.S. more competitive with countries such as China.

Even if the parties agreed on the size and composition of an infrastructure bill, however, they are far apart on how to pay for it. Republicans have been more interested in user fees such as taxes on heavy trucks and repurposing unused money while the Biden administration is pushing a corporate tax increase that would reverse much of the 2017 tax law.

Republican leaders, including Sen. Mitch McConnell (R., Ky.) have declared those 2017 tax cuts—the biggest domestic-policy legislative achievement of the Trump administration—a red line.

But user fees could clash with Mr. Biden’s promise to avoid raising taxes on households making under $400,000, and key Democrats already have expressed their opposition.

“The suggestion is somehow middle class workers are supposed to pay what mega-corporations will not,” Sen. Ron Wyden (D., Ore.), chairman of the Finance Committee, said at a hearing last week.

Democrats and Republicans are also at loggerheads over plans to establish an independent commission to investigate the Jan. 6 attack on the Capitol. The House passed a bill Wednesday to create the panel, but Republicans are objecting.

An initial Senate procedural vote on the commission-creating bill could occur as early as this week. If that fails, Democrats could empower congressional committees that they control

“We should be coming together in a bipartisan way to do a thorough investigation to make sure that the second time in American history that our Capitol was taken is the last time,” Sen. Cory Booker (D., N.J.) said on CNN.

That measure got 35 Republican votes in the House, but GOP leaders and senators have called the panel unnecessary and duplicative of law-enforcement efforts.

Several congressional committees already are investigating the government’s planning and response to the riot and attack, and the internal watchdogs of four government agencies also have launched reviews of officials’ actions. In addition, the Justice Department’s criminal probe has led to charges against more than 400 people of offenses stemming from the Jan. 6 attack.

Ms. Collins said she supports a commission but wants to make sure its staff is hired in a bipartisan way and that it finishes its work this year. She said she was optimistic that those two issues could be resolved, citing a recent conversation with House Speaker Nancy Pelosi (D., Calif.)

Sen. Roy Blunt (R., Mo.) said it was too early to create a commission and that there was already bipartisan committee work happening.

“Is the priority to secure the Capitol, to do what we need to do to better train, better prepare Capitol police, decide what we want to do in the future,” he said on Fox News Sunday. “Or is the priority to take what will be a couple years, in my view, to decide what happened inside the White House?”

—Sabrina Siddiqui contributed to this article.

Write to Richard Rubin at richard.rubin@wsj.com