For decades, senators looking to push major budget and tax legislation through Congress on a simple majority vote have had to win the blessing of a single unelected figure on Capitol Hill.

The Senate parliamentarian, a civil servant who acts as the arbiter and enforcer of the chamber’s byzantine rules, has traditionally been in a position to make or break entire presidential agendas. That includes determining whether budget and tax legislation can be fast-tracked through Congress and shielded from a filibuster, allowing it to pass along party lines through a process known as reconciliation.

Now, in their zeal to deliver President Trump’s domestic policy agenda in “one big beautiful bill” of spending and tax cuts, Senate Republicans are trying to steer around the parliamentarian, busting a substantial congressional norm in the process.

The strategy would allow them to avoid getting a formal thumbs up or thumbs down on their claim that extending the tax cuts that Mr. Trump signed into law in 2017 would cost nothing — a gimmick that would make it easier for them cram as many tax reductions as possible into their bill without appearing to balloon the deficit.

In recent days, all eyes have been on Elizabeth MacDonough, the parliamentarian, to see whether she would bless the trick, smoothing the path for the G.O.P. bill. But on Wednesday, Republicans signaled that they planned to take extraordinary action to go around her altogether.

Rather than have Ms. MacDonough weigh in, they asserted that Senator Lindsey Graham of South Carolina, as chairman of the Budget Committee, could unilaterally decide the cost of the legislation, citing a 1974 budget law. Senate Republicans on Wednesday unveiled a new budget resolution they planned to put to a vote as early as this week. And Mr. Graham declared in a statement that he considered an extension of the 2017 tax cuts to be cost-free.