President Joe Biden has agreed to a faster phase-out of the $1,400 stimulus checks in the next relief package, according to two Democratic sources.Some moderate Democrats in the Senate have been urging Biden and Democratic leaders to further target the direct payments.

According to the sources, individuals making over $80,000 a year will no longer qualify for the payments — the cap in the House bill was $100,000. The limit for heads of households will be $120,000 a year, compared to $150,000 in the House bill. The ceiling for couples will now be $160,000 a year, versus $200,000.

The new proposal begins phasing out the checks the same income levels as the House bill — $75,000 for individuals, $112,500 for heads of households and $150,000 for joint filers.

Sen. Debbie Stabenow (D., Mich.) told reporters the agreement was a "reasonable compromise."

"I think we're really in a good spot and, frankly, the most important thing is to get this done," said Stabenow.

Moderate Democrats have also been pushing to lower the enhanced unemployment benefits to $300 per week, down from $400 per week. The House bill extends the benefits through Aug. 29, but progressives like Senate Finance Chairman Ron Wyden (D., Ore.) have been fighting to extend them through September.

Two Democratic sources said the Senate bill is expected to keep the benefits at $400 per week through the end of August.

"Adding $400 per week to jobless benefits and covering gig workers and the self-employed is the boldest action Congress has ever taken to support jobless Americans during an economic crisis," said Wyden in a statement to Yahoo Finance. "I pushed hard for keeping that sixth month of benefits and am going to fight like hell to extend them in August."

Wyden said he'll also continue to fight for long-term unemployment insurance reform and automatic stabilizers in the months ahead.

Progressives in the House — who are already upset that the minimum wage increase is not allowed in the package — have urged the Senate not to alter the relief package too much. The House will have to vote on any changes the Senate makes.

In an interview with Yahoo Finance Live, Rep. Pramila Jayapal (D., Wash.) warned against tightening eligibility for stimulus checks or decreasing the weekly unemployment boost.

"It will be a real problem if this gets undermined in the Senate," said Jayapal.

In the White House press briefing, Press Secretary Jen Psaki told reporters the President does not want to lower the thresholds to start phasing out the checks — but he's open to phasing the payments out at a faster pace. Psaki said Biden is "comfortable" with where negotiations stand.

Jessica Smith is chief political correspondent for Yahoo Finance, based in Washington, D.C. Follow her on Twitter at @JessicaASmith8.

Read more:

• Lawmakers urge Democratic leaders to waive taxes on jobless benefits in COVID relief bill

• Biden picks Katherine Tai for US trade representative: Reports

• Key lawmaker 'optimistic' enhanced jobless aid won't be needed past August

• Rep. Axne: Robinhood, trading platforms have responsibility to protect investors

• Why this bitcoin-owning senator wants Elon Musk to move to Wyoming