Sen. John Cornyn John CornynCalls grow for Biden to expand election map in final sprint Bipartisan praise pours in after Ginsburg's death Chamber of Commerce endorses McSally for reelection MORE (R-Texas) said on Monday that Senate Republicans want to roll out a new draft of their bill to repeal and replace ObamaCare as soon as this week, with a vote next week.

"We're just trying to get a good picture of what the alternatives are, and hopefully next week we'll be prepared to take the bill up and vote on it," the No. 2 Senate Republican told reporters.

He added that Republicans are still waiting to hear back from the Congressional Budget Office (CBO) but could release an updated draft of their bill by the end of the week.

A Senate GOP aide said senators are expected to get briefed on revisions that have been made to the bill at Tuesday's lunch, and revised text of the bill could be released later this week.

Sen. Pat Toomey (R-Pa.) earlier on Monday told CNBC he believed that a bill might be floated as early as Monday.

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Cornyn's comments come as Senate GOP leadership scrambles to figure out a way forward on its bill to repeal and replace ObamaCare. The Texas Republican pushed back against suggestions that they cancel the August recess or delay leaving town at the end of July.

"I'm for getting our work done now. I don't think stringing it out any longer than next week helps us with the product," he said. "My personal goal is to have it up on the floor ready to vote, have the vote-a-rama, next week."

Republicans are returning to Washington Monday increasingly pessimistic about their chances to pass their repeal bill.

Cornyn, who met with Senate Majority Leader Mitch McConnell Addison (Mitch) Mitchell McConnellMomentum growing among Republicans for Supreme Court vote before Election Day Trump expects to nominate woman to replace Ginsburg next week Video of Lindsey Graham arguing against nominating a Supreme Court justice in an election year goes viral MORE (R-Ky.) on Monday afternoon, said Republicans have been in "continuous" discussions across the recess.

"It's going to be ... a test for all of us. But I'm hopeful and I'm optimistic that we'll pass the test," he said.

Senate Republicans sent two proposals to the CBO late last month, one including an amendment from Sen. Ted Cruz Rafael (Ted) Edward CruzVideo of Lindsey Graham arguing against nominating a Supreme Court justice in an election year goes viral Sunday shows preview: Justice Ginsburg dies, sparking partisan battle over vacancy before election Democrat on Graham video urging people to 'use my words against me': 'Done' MORE (R-Texas) and one without.

The Cruz proposal would give insurance companies the freedom to sell any kinds of health plans they want as long as they also sell at least one plan that qualifies under the regulatory requirements of the Affordable Care Act.

But the amendment has drawn pushback from aides and other GOP senators, who warn that the proposal won't be able to get enough support to pass.

Cornyn added on Monday that Republicans want to get "some basic information" from the CBO on the proposal.

"Everybody's concerns need to be taken into account," he said. "The goal is, though, once you secure support from one person, you don't want to lose it from another. So you can see the challenge."

With a slim 52-seat majority, they can only afford to lose two GOP senators and still let Vice President Pence break a tie. Roughy 10 GOP senators have come out against the current version of the Senate healthcare bill.

— Peter Sullivan contributed.