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The Treasury Department will ask Congress for $500 billion in direct payouts for taxpayers as part of a $1 trillion stimulus package to combat the economic effects of the coronavirus pandemic, according to a proposal obtained by NBC News.

The two rounds of direct payments to taxpayers, each totaling $250 billion, would be sent on April 6 and on May 18, according to the proposal. They would be tiered payments, with the amounts based on income level and family size, the proposal says. Both payments would be for the same amount.

President Donald Trump was asked about the proposal at a coronavirus briefing at the White House on Wednesday. "I don't want to get in that right now" because "different numbers" are being discussed, he said, but he added, "We want to go big."

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The Treasury is also asking for $50 billion for the airline industry, a small business interruption loan program of $300 billion and $150 billion for other distressed sectors, the proposal said.

Trump had initially favored a payroll tax cut to bring relief to Americans struggling financially, but a number of lawmakers across party lines have been pushing for direct payments to Americans.

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The White House first floated its support for sending checks directly to Americans during a news conference Tuesday. Treasury Secretary Steven Mnuchin said Trump wanted Americans to get relief as soon as possible.

"Americans need cash now, and the president wants to get cash now," Mnuchin said Tuesday. "And I mean now, in the next two weeks."

Trump is pushing the Senate to act quickly. "The president has a pretty specific goal on all of us by suggesting that the checks go out by April 6th," said Sen. Kevin Cramer, R-N.D.

The disbursements are likely to be targeted based on means and family size, Sen. Josh Hawley, R-Mo., and a Senate aide told NBC News.

Sen. John Cornyn, R-Texas, told reporters the direct payments were being considered for a third set of coronavirus relief measures, which the Senate is planning to tackle after voting Wednesday to pass the second package, which includes provisions for free coronavirus testing, food assistance and unemployment protections.

"The third piece will be composed, I think, of direct payments to those individuals who are not getting a paycheck now," Cornyn said. "We're trying to figure out what the fastest way [is] to get that money to them to get them over the hump during this period during which their employer shut down, and as well as what do we need to do to keep the airplanes flying and the hotel industry that employs an awful lot of people in keeping their viability."

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On Monday, Sen. Mitt Romney, R-Utah, proposed giving every adult $1,000 to help meet financial obligations. A group of Senate Democrats, led by Michael Bennet of Colorado, Cory Booker of New Jersey and Sherrod Brown of Ohio, proposed sending as much as $4,500 to each American.

Not every lawmaker is on board.

Sen. Lindsey Graham, R-S.C., said he wants the stimulus to focus on ensuring that small businesses can continue to pay their employees.

"I'm not looking to send people a government check. I'm looking to keep their paychecks coming. And if we don't do that, we're making a huge mistake," Graham told reporters Wednesday, adding that $1,000 checks wouldn't really get "you anywhere, because there's no place to spend a thousand dollars."