Top Senate Republicans are predicting that they'll get enough GOP support to help advance a deal on the debt ceiling as Senate Minority Leader Mitch McConnell (R-Ky.) pitches his caucus on the agreement.

Under the deal, the House and Senate will first need to pass a bill that sets up a subsequent one-time simple majority vote to raise the debt ceiling.

Though Democrats would then be able to raise the debt ceiling on their own, they would need 10 GOP votes to advance the initial legislation that lays out how the debt ceiling vote will work.

The House introduced the bill Wednesday that sets up and lays out the instructions for the debt ceiling vote and could vote to approve it as soon as Wednesday.

"The majority party has to raise the votes to raise the debt limit, and the Democrats know that, and they're willing to do it. ... We think that's a perfectly appropriate way to handle this," said Sen. John Thune (S.D.), the No. 2 Senate Republican.

Thune added that he believed they would be able to get 10 GOP votes for the deal.

Asked if he expected 10 Republicans would support the deal, Sen. Roy Blunt (Mo.), the No. 4 Senate Republican, said, "I would expect that's what happens. It happened last time and maybe more."

GOP leaders view the agreement as a win for them because it's expedited, which is what they want, and forces Democrats to raise the debt ceiling on their own to a specific number. A quicker process also lets them spend more time attacking Democrats over their climate and social spending plan.

An aide predicted that a "vast majority" of Republicans would be supportive of the agreement, though they acknowledged that it didn't mean that many would vote for the agreement setting up the debt ceiling vote on the Senate floor. That vote, according to aides, could happen in the Senate as soon as this week.

The deal comes after weeks of negotiations over the debt ceiling between McConnell and Senate Majority Leader Charles Schumer (D-N.Y.). Republicans had offered to speed up Democrats' passage of the debt ceiling on their own under reconciliation, a budget process, but aides argued that the deal rolled out on Tuesday accomplishes the same goal.

Senate leaders had discussed tying the debt hike to a sweeping defense bill but dropped that plan amid pushback from House Minority Leader Kevin McCarthy (R-Calif.) and Senate Republicans.

Instead, the deal to set up a separate debt ceiling vote is expected to be folded into legislation on preventing Medicare cuts.

Schumer indicated earlier this week that talks with McConnell were going well.

"We will also work to address the debt limit and preserve the full faith and credit of the United States, and I want to thank Leader McConnell for his cooperation in that regard," he said.