But some Republicans immediately resisted the idea, leading to no apparent progress after a meeting with Treasury Secretary Steven Mnuchin and White House chief of staff Mark Meadows.
"There’s no consensus on anything," Sen. John Cornyn, R-Texas, said after the meeting. "Just a lot of expression on views."
Cornyn was one of several GOP senators who said they were opposed to a short-term extension.
Another was Sen. David Perdue, R-Ga., a Trump ally who faces a competitive re-election race.
"I do not" support a short-term jobless insurance extension, Perdue said. "I'd rather find other ways to incentivize through a tax credit or whatever incentive to get people back to work ... What we've done is we've given people an incentive not to come back to work and I think that's wrong."
A federal moratorium on evictions lapsed Friday. An unemployment bonus of $600 per week expires July 31. As of midweek, the prospects of a deal in Congress by that deadline were close to nonexistent.
"We want to work on the evictions so that people don’t get evicted. We’ll work on the payments for the people. And the rest of it, we’re so far apart, we don’t care," Trump told reporters at the White House. "We’re focused on those two things. We want to take care of them now. The rest, we can discuss later."
The new push comes as the Senate Republicans fracture over their $1 trillion plan, with some opposing new spending on virus aid as others want more help for their struggling states. The plan would be a counteroffer to the Democrats' $3.4 trillion package that passed the House in May, which includes a six-month extension of $600-per-week jobless benefits and a ban on evictions for renters.
The negotiations have gone nowhere as Democratic leaders reject the Senate plan.
Mnuchin, speaking alongside Trump on Wednesday, said the president "is very focused on evictions and unemployment" and "wants to look at giving us more time to negotiate this."
Sen. Marco Rubio, R-Fla., whose state set a record with more than 200 daily COVID-19 deaths Tuesday, said a short-term bill "may need to happen."
"Ultimately, it's clear we're not going to have a universal agreement in place by Friday," he said.
One senator in the room said Mnuchin didn't make a pitch to the Republicans for a short-term unemployment extension or an eviction moratorium, but rather discussed the situation and sought their ideas. The senator, who requested anonymity to speak candidly about a private meeting, was skeptical of a small bill and suggested Trump may have floated it without much of a commitment.
Sen. Mike Rounds, R-S.D., said he doesn't support a short-term aid bill, nor does he believe it's the most likely solution. He blamed the Democrats, suggesting that "they may not be wanting to do anything at this stage of the game."
Leaving the meeting, Meadows said he had "no updates" on progress for a short-term measure.
Senate Finance Committee Chair Chuck Grassley, R-Iowa, said Republicans "sure don't want to" resort to a short-term measure but they are up against some deadlines.
"One would be unemployment ending, and obviously you know we want to continue that," he said. "The other thing is the evictions are going to — the prohibition on evictions is going to end. And that’s going to be a hardship for a lot of people. I think we need to get that stuff done right away and get the crisis behind us, so we have time to think of other things that need to be done."
House Speaker Nancy Pelosi, D-Calif., has rejected a short-term fix and insisted on a broader solution. She reiterated her opposition Wednesday when asked what she makes of Mnuchin talking about a patch for unemployment benefits and an eviction moratorium.
"Nothing," she said, according to a pool reporter. "Not even 'not much' — nothing."
"So we don't know why the Republicans come around here with a skinny bill that does nothing to address really what's happening with the virus, and has little, little of this and a little of that," Pelosi said. "We're not accepting that."
After another day meeting with Pelosi and Sen. Chuck Schumer, D-N.Y., Mnuchin said the sides "don’t have an agreement on anything."
Sen. John Kennedy, R-La., said he was open to Trump's idea.
"At this juncture, that makes sense to me," he said. "Because I think it's going to be difficult in the short term to get agreement on a larger package, and people need help."
Kennedy said part of the problem with the larger package was that the White House was insisting on unrelated spending provisions, such as $1.75 billion that Trump wants for a new FBI building.
"I asked the powers that be to take the spending porn out of the bills," he said.