WASHINGTON — The Senate Judiciary Committee could vote as soon as next week on bipartisan legislation introduced Wednesday that would allow special counsels such as Robert Mueller to appeal their firing to a panel of judges and possibly be reinstated.

The committee’s chairman, Senator Charles E. Grassley, Republican of Iowa, has asked its top Democrat, Senator Dianne Feinstein of California, to give the legislation a procedural signoff that would allow an expedited formal drafting and vote — most likely on April 19.

The chairman’s request came as many Republicans continue to say that no legislative action is necessary, despite continuing threats from President Trump against Mueller and senior Justice Department officials. Republican leaders have steadfastly maintained that Trump knows the consequences of firing Mueller too well to do so.

But Republicans are under pressure to shift their stance. Even if the legislation never passes Congress, a bipartisan committee vote would send a signal to Trump and push Republican leaders to respond.

The compromise bill, written by Republican Senators Thom Tillis of North Carolina and Lindsey Graham of South Carolina and Democrats Chris Coons of Delaware and Cory Booker of New Jersey, would codify an existing Justice Department regulation that says a special counsel may be fired only by the attorney general, and only for good cause, such as misconduct.

The bill also creates a 10-day window within which a special counsel can seek judicial review of the firing. If it is determined that the special counsel was not, in fact, fired for good cause, then he or she would be reinstated.

Democrats, who have been clamoring for Congress to act to protect Mueller, touted the new legislation as a breakthrough but privately conceded that odds remained stacked against its passage. Tillis, in his own comments, sought to put distance between the measure and the bubbling tensions between Trump and Mueller’s investigation into Russia’s election interference and possible ties to the Trump campaign.

“This compromise bipartisan bill helps ensure that special counsels — present or future — have the independence they need to conduct fair and impartial investigations,’’ Tillis said. “The integrity and independence of special counsel investigations are vital to reaffirming the American people’s confidence in our nation’s rule of law.’’

Grassley, who has raised concerns about the constitutionality of the bill, plans to offer an amendment to it that would formally require the Justice Department to produce reports to Congress each time there is a change in scope to a special counsel’s investigation or if he or she is fired. The department would also have to prepare a detailed final report about what a special counsel found and explain decisions to charge or decline to charge particular suspects, according to Republican committee aides.

The aides said that Grassley is studying statutes that place some limits on the firing of certain agency inspectors general, with an eye toward echoing some of that language for a Justice Department special counsel. Some statutes, for example, require the executive branch to give Congress 30 days’ notice and an explanation before such an official is removed.

With or without Grassley’s amendment, even proponents of the bill concede that its chances of becoming law remain slim. Senator Mitch McConnell, Republican of Kentucky and the majority leader, said Tuesday he had not seen “clear indication yet’’ that such a bill was necessary.

Speaker Paul Ryan, Republican of Wisconsin, has not yet commented on the new bill. If it passed Congress, Trump would most likely veto it.

Still, a bipartisan committee vote would be significant.

Senators from both parties introduced a pair of bills late last summer aimed at protecting Mueller, and the Judiciary Committee held a hearing to consider their constitutionality. But an effort to combine the two bills languished for months, as urgency among Republicans dissipated.

Lawmakers involved in drafting the new bill said the pace began to pick up again as Trump’s increasingly aggressive posture against Mueller prompted speculation that he might try to fire him.