Instead, he said he would consider targeting them differently than the previous two rounds of direct aid to Americans. Members of both political parties have questioned whether the $1,400 payments he has proposed would go to people who don’t need the aid.

“We can better target that number. I’m OK with that,” Mr. Biden said, according to the people.

White House press secretary

Jen Psaki

said later Wednesday that Mr. Biden is open to changes in the threshold for who would qualify for the $1,400 stimulus checks.”That’s something that has been under discussion,” she said.

Mr. Biden also said he was flexible on the overall cost of the package, which Democrats have started advancing through Congress through a process that will allow them to pass it along party lines, according to the people familiar with the call. He said Democrats could make “compromises” on several programs in the proposal, one of the people said.

Ms. Psaki said Mr. Biden isn’t expecting the final package to look exactly like what he proposed. “He knows that that’s part of the legislative process,” she said.

Beyond sending money to many Americans, the $1.9 trillion proposal would direct aid to state and local governments, provide funds for distributing Covid-19 vaccines and enhance federal unemployment benefits. Money would go toward schools, child-care facilities and renters under the plan, which also seeks to raise the federal minimum wage to $15 an hour.

Republicans have called Mr. Biden’s plan too expensive and premature after Congress approved roughly $900 billion in aid in December, and they have criticized provisions like raising the minimum wage as unrelated to the pandemic. A proposal advanced by 10 Senate Republicans would provide $618 billion in relief, paring back Mr. Biden’s proposals on unemployment insurance and direct checks and eliminating others.

In meetings with Democrats, Mr. Biden has said the GOP plan is too small to deal with the effects of the pandemic.

Senate Minority Leader Mitch McConnell (R., Ky.) said Democrats “seem desperate to make their first act in power the same kind of massive, partisan, poorly targeted borrowing spree that permanently wounded the last Democratic presidency right out of the gate.”

Democrats are divided on who should benefit from a new round of direct payments to Americans. Previous relief bills began phasing out the payments for people making more than $75,000 a year and married couples with incomes of more than $150,000. The Biden administration hasn’t yet detailed the income cutoffs it would put in place, though some Democrats have said they want to stick with the same cutoffs as the previous efforts.

Other Democrats see the current thresholds as too generous, allowing Americans who haven’t been economically harmed during the pandemic to receive government aid.

Speaking to reporters at the White House on Wednesday after a meeting with Mr. Biden, Sen. Chris Coons (D., Del.) said: “We did have a conversation about the direct payments and how those might be modified in a way to ensure they’re targeted.” He added that Mr. Biden is “not going to forget the middle class.”

The Republican plan would reduce the size of the checks to $1,000 per adult and start to phase out the payments for individuals who make $40,000 a year or more and married couples with incomes of $80,000 or more. A bipartisan group of senators involved in kickstarting the last coronavirus relief bill also has discussed how to target the relief checks.

Democrats this week began pushing forward with a process called reconciliation, which would allow them to pass the coronavirus relief bill with fewer than the 60 votes required for most legislation in the Senate. With the Senate split 50-50—Vice President

Kamala Harris

can break ties—Democrats cannot afford to lose a single vote on the package in the Senate.

According to a Penn-Wharton Budget Model estimate, households in the short term would save about 73% of the money they receive from the direct payments if Mr. Biden’s proposal for $1,400 per person uses the same income thresholds as earlier payments. That savings figure includes paying down debt.

Checks more focused on those who lost income would be more likely to be spent, the group said.

“A large portion of people getting checks are people who are just going to save it because they’re not in these industries who are being hurt,” said Rich Prisinzano, the group’s director of policy analysis.

Proponents of sending direct payments argue that casting a wide net helps people who may be slipping through the cracks of other aid programs.

Mr. Biden met with another group of Democratic senators in the Oval Office on Wednesday. After the meeting, Senate Majority Leader Chuck Schumer (D., N.Y.) told reporters it was a substantive discussion, and that Democrats were united on passing a large package.

“We want to do it bipartisan, but we must be strong,” Mr. Schumer said. “We cannot dawdle, we cannot delay, we cannot dilute, because the troubles that this nation has and the opportunities that we can bring them are so large.”

The meetings on Wednesday are the latest in a flurry of meetings the new president has had with lawmakers on Capitol Hill. He spoke with Senate Democrats on Tuesday, urging lawmakers to adopt a large package. On Monday, Mr. Biden hosted the group of 10 Senate Republicans at the White House to discuss their $618 billion alternative plan.

—Tarini Parti and Richard Rubin contributed to this article.

Write to Andrew Duehren at andrew.duehren@wsj.com and Eliza Collins at eliza.collins@wsj.com.