It comes as Mr Biden, inaugurated last week, faces increasing problems in his effort to win bipartisan backing for the initial legislative effort of his presidency.
Politicians on the right question the wisdom of racking up bigger deficits while those on the left are urging Mr Biden not to spend too much time on bipartisanship when the pandemic is killing thousands of Americans each day.
At least a dozen senators met virtually for more than an hour with White House National Economic Council director Brian Deese and other senior White House officials on Sunday.
Many hope to approve a relief package before former president Donald Trump’s trial, which is set to begin in two weeks, overtakes Washington’s attention.
Senator Angus King, an independent from Maine, called the opening talks a “serious effort”.
“There was not a hint of cynicism or lack of commitment to at least trying to work something out,” he said.
The White House did not seem to budge on breaking up the package or reducing the overall price tag, even as it pushes for bipartisan support.
There was also no discussion of pushing it through with a procedural move that could be done without Republicans, Mr King said.
Senators from both parties raised questions about the economic aid provisions, particularly making direct 1,400 dollar (£1,020) payments to Americans more tailored to recipients based on need.
They also wanted more data on how the White House reached the 1.9 trillion dollar figure.
Many of the senators are from a bipartisan group that struck the contours of the last Covid-19 deal approved late last year.
Out of the gate, Mr Biden has made clear that quickly passing another round of coronavirus relief is a top priority as he seeks to get the surging pandemic and the related economic crisis under control, while demonstrating he can break the gridlock that has ailed Congress for much of the last two presidencies.
Mr Biden and his aides have stressed that his plan is a starting point and that finding common ground on relief should be attainable considering the devastating impact the pandemic is exacting on Democratic and Republican states alike.
With more than 412,000 dead and the economy again losing jobs, Mr Biden has argued there is no time to lose.
“We’re going to continue to push because we can’t wait,” said White House principal deputy press secretary Karine Jean-Pierre.
“Just because Washington has been gridlocked before doesn’t mean it needs to continue to be gridlocked.”