A bipartisan budget agreement overwhelmingly approved by the House appears likely to survive any procedural challenges and ultimately pass the Senate, according to interviews with key Republican senators.

The House voted 332 to 94 on Thursday night to approve a two-year budget outline, and Senate debate is expected to begin Tuesday, with a final vote expected later next week. While the measure will need just a majority of senators to pass, it first must clear a procedural vote requiring at least 60 votes in support. The 55 members of the Senate Democratic caucus who have signaled general support for the plan will need at least five Republican senators to join them.

Four Republican senators — John McCain (Ariz.), Jeff Flake (Ariz.), Susan Collins (Maine) and Richard Burr (N.C.) — signaled Friday that they would vote to end debate and proceed to final passage of the budget agreement. Of them, only McCain said he is “very likely” to vote for the actual budget bill in hopes of avoiding another government shutdown.

“It would be outrageous for us to shut down the government again and I’m not going to do that to my constituents,” McCain told reporters.

Flake said he was “leaning no” on the budget bill, as did Burr. Collins said she is still reviewing the legislation.

Other Republicans, including Charles E. Grassley (Iowa) and Lisa Murkowski (Alaska), said they were considering voting to end debate on the bill even if they ultimately vote against it.

Two other Republican senators who often vote with Democrats to at least end debate on high-profile legislation — Sens. Kelly Ayotte (R-N.H.) and Lindsey O. Graham (R-S.C.) — said late Thursday that they would vote against the budget bill. Both of them cited concerns that the measure fails to properly pay down the federal debt and pays for government operations in part by requiring military retirees to endure lower cost of living increases on their retirement benefits.

Their decision is surprising in part because Ayotte and Graham held campaign-style events last year with McCain in military communities in hopes of pushing Congress to restore funding for the Pentagon that was set to suffer billions of dollars in losses next year from mandated automatic budget cuts, known as “sequestration.” The new budget proposal restores billions of dollars to the defense budget.