Senate Republicans are making progress on coronavirus relief legislation with daily talks with the White House and hope to put forth a narrow plan as soon as next week, according to the White House and one GOP leader.

"I expect them to pass a bill, or at least put forth a bill in hopes of getting to that 60-vote threshold sometime next week," White House chief of staff Mark Meadows told CNBC Tuesday.

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And Sen. John Barrasso, R-Wy., remarked at the Capitol Tuesday morning that Senate Republicans would like to move their "focused, targeted" plan for COVID-19 relief next week.

"That's the goal," Barrasso told PBS NewsHour correspondent Lisa Desjardins.

The Senate plan would be tailored to getting people back to work and kids back to school and targeting the coronavirus disease. The Republicans will "leave out" the "many things that [House Speaker] Nancy Pelosi has put in her bill that are unrelated to coronavirus," Barrasso said.

The Senate has been out on recess during August, but the GOP caucus has been holding daily calls with Meadows and Treasury Secretary Steven Mnuchin, according to a source familiar. The group was expected to talk again Tuesday on charting a path forward.

A spokesperson for Senate Majority Leader Mitch McConnell, R-Ky., did not have any immediate scheduling announcements for next week.

Republicans have yet to forge any consensus among their own, with some members concerned about any new spending. A source told Fox News they still have not found an agreement but are working on it.

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Any plan would need 60 votes to pass the Senate, which means it would require help from Democrats. Once the GOP gets support from its caucus, they would move a bill, Fox News is told.

House Democrats passed the HEROES Act, a more than $3 trillion stimulus plan, in May and have been pressuring McConnell to pass something in the Senate ever since.

Meadows and Mnuchin had been in talks with Pelosi, D-Calif., and Senate Minority Leader Chuck Schumer, D-N.Y., but their efforts haven't produced a compromise. Democrats accuse the White House of not understanding the gravity of the economic crisis and health emergency.

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Meadows on Tuesday said Democrats' push for nearly $1 trillion in aid to state and local governments hard-hit by coronavirus revenue losses is not based "in reality." He said Republicans would be willing to support $150 billion in new funding and give flexibility to how states spend the previous $150 billion that was already passed.

The Senate is expected to return to Washington after the Labor Day holiday.

Fox News' Chad Pergram contributed to this report.