The U.S. government is already in the process of sending $600 stimulus checks to millions of American households. Now, the Senate is debating whether to approve larger checks after President Donald Trump demanded last week – and then again Wednesday morning – that the direct payments be sized up to $2,000.

The sprawling relief package signed by Trump on Sunday night provides $600 to adults with annual incomes up to $75,000, plus another $600 per child. A family of four could receive $2,400 if it meets the income requirements.

Under that bipartisan stimulus law, the payment amount decreases by 5 percent of every dollar over the “full stimulus” limit. That means that checks under $600 will still go out to individuals making up to $87,000 a year.

A new law passed by the House this week would increase the full stimulus check amount to $2,000 (or $1,400 on top of the $600 already approved, for most people). The phase-out structure is still the same, but because the check itself is larger, people making up to $115,000 would still get something. The new bill also covers adult dependents.

Marc Goldwein, senior policy director at the Committee for a Responsible Federal Budget, broke down who would get payments under the House plan, and how large those deposits would be.

For example, under the stimulus package already signed into law, a family of four making more than $198,000 would get nothing. Under the House plan, the high-income threshold rises to $310,000. In that plan, a family of four making up to $150,000 would receive $8,000. If that same family were making $300,000, they would still get $500.

Ernie Tedeschi, an economist and former Treasury Department official in the Obama administration, estimated that 8.8 million more families will get a check of some sort under the House plan.

That comes with a price tag. The Joint Committee on Taxation estimates that approving stimulus checks of $2,000 would cost $464 billion. That would be on top of the $900 billion package that includes expanded unemployment benefits, aid for small businesses, vaccine distribution and more.

Congressional Republicans had long sought to keep the price tag of a relief bill under $1 trillion. But given Trump’s steadfast insistence on $2,000 checks, pressure is mounting on Senate Majority Leader Mitch McConnell, R-Ky., and the GOP to give the green light for larger checks.

On Tuesday, McConnell blocked initial consideration of the House bill. On one side of the divide within the Republican party, an increasing number of GOP Senators have shown support for $2,000 checks. Trump on Tuesday tweeted that “unless Republicans have a death wish, and it is also the right thing to do, they must approve the $2000 payments ASAP.”

Meanwhile, there are still plenty of GOP lawmakers wary about extending money to families with higher incomes, particularly those who are not bearing the brunt of the covid crisis.

Sen. Susan Collins, R-Maine, told reporters on Tuesday told reporters that she was concerned about how the House bill was structured.

“For example, for a family of four, the income level at which you would get nothing is more than $300,000,” Collins said. “And in a state like mine, that’s high income. So I don’t know whether it’s possible to put a cap on it or make some changes.”

On Tuesday night, the Treasury Department said it had started issuing the first round of $600 stimulus payments and that some Americans could begin receiving them this week. The payments will be distributed automatically, with no action required for people who qualify.

Treasury also said that if another bill was enacted to cover larger checks, payments that were already issued “will be topped up as quickly as possible.”

The Washington Post’s Tony Romm and Mike DeBonis contributed to this article.