President Donald Trump's call for "higher numbers" amid an impasse over a new coronavirus stimulus has upended a GOP effort to unite against Democrats pushing for a larger aid package.

Trump said on Wednesday, "Go for the much higher numbers, Republicans," in a tweet referencing coronavirus stimulus payments -- a message that stands to undercut the strategy Hill Republicans have pursued of opposing Democratic efforts to negotiate a higher top-line number than GOP lawmakers have so far put forward.

The President's message privately frustrated Republicans, who wanted the focus to be on House Democratic infighting over their party's response to the pandemic and on the refusal of Democratic leadership to support a narrower, pared-back relief effort. Instead, the tweet immediately put a spotlight on Republicans and prompted questions over whether they would join Trump's call for a larger price tag.

Top Republicans have so far not embraced the call from the President. Some GOP lawmakers pushed back on Trump's suggestion to go for "higher numbers" on Wednesday, while others dodged questions about whether they would support such a move.

Senate Majority Whip John Thune of South Dakota, reacting to Trump's tweet urging the GOP to get behind more stimulus money, warned that Republican support could deteriorate in the face of a more expensive proposal.

Asked if Republicans would support higher numbers, Thune responded, "I'm not sure what higher number means ... but I know sort of what the threshold is for what we can (get) Republican votes for in the Senate, and if the numbers are too high anything that got passed in the Senate would be passed mostly with Democrat votes and a handful of Republicans. It's got to have a realistic range if we want to maximize, optimize, the number of Republican senators that will vote for it."

At the end of July, Senate Republicans unveiled a roughly $1 trillion stimulus proposal, which Democrats criticized as inadequate as they remained unified around a more costly $3 trillion proposal of their own.

Since then, Democrats have offered to drop their top-line demand to $2.2 trillion, but the White House and Senate Republicans have rejected that. Instead, Senate Republicans attempted unsuccessfully to pass a scaled-back, roughly $500 billion proposal last week, a measure that Democrats blocked.

Asked what the highest number is that Republicans would support, Thune said on Wednesday, "I mean initially it was a trillion dollars, and we had a lot of resistance to that, so I think our members are in that range, probably," adding, "but as you go upwards from there you start losing Republican support pretty quickly."

House GOP leader Kevin McCarthy of California dodged when pressed by reporters about Trump's tweet. When asked if Republicans would support a higher number than they have offered so far, he deflected the question by instead accusing Democrats of delays. "I don't think the Democrats actually want one. I think the speaker delays with this," he said and then walked into the House chamber.

House GOP Conference Chairwoman Liz Cheney of Wyoming said on Wednesday, "I would be hesitant to make a commitment about a level that is that high," adding, "I'm really concerned about funds that have been appropriated and haven't been spent yet. I think the real solution to this is we've got to get the economy going again."

Senate Minority Leader Chuck Schumer of New York and House Speaker Nancy Pelosi of California seized on Trump's tweet Wednesday.

"We are encouraged that after months of the Senate Republicans insisting on shortchanging the massive needs of the American people, President Trump is now calling on Republicans to 'go for the much higher numbers' in the next coronavirus relief package," the two top Democrats in Congress said in a statement.

"We look forward to hearing from the President's negotiators that they will finally meet us halfway with a bill that is equal to the massive health and economic crises gripping our nation," they said.

Asked to clarify exactly what the President was calling for in his tweet, White House press secretary Kayleigh McEnany said during a briefing on Wednesday, "What the President was referring to was the $500 billion bill that passed the Senate. ... It didn't include direct payments. So he wants more than the $500 billion, and he is very keen to see these direct stimulus payments and we hope that Nancy Pelosi will work with us in good faith."

When asked about Trump's tweet urging Republicans to back more stimulus money, Sen. Roy Blunt of Missouri, a member of the Republican Senate leadership team, said he thinks "the number is going to be higher than our trillion dollars."

"I think there's a deal to be had here," Blunt said. "My concern is that the window probably closed at the end of this month and we need to get busy finding out what we can all agree on. And I think the number is going to be higher than our trillion dollars."

GOP Sen. Lindsey Graham of South Carolina said on Wednesday to "count me in the camp of trying to get something done" following the President's tweet, but added that he believes "we can do this thing well below $2 trillion and meet the needs."

CNN's Kristin Wilson contributed to this report.