Senate Republicans are coming under intense pressure from President Trump and right-wing colleagues to embark on an old-fashioned filibuster fight in an effort to ram through a voter identification bill that their party regards as crucial to salvaging their dimming chances of winning the midterm elections.

The move, which Senator John Thune, the South Dakota Republican and majority leader, has been reluctant to undertake, involves using tactics that haven’t been employed for decades, and could paralyze the Senate indefinitely with no guarantee of success.

But with 50 Republican senators now officially behind the legislation, its backers are escalating calls for their party to force Democrats who have promised to block the bill to wage a so-called talking filibuster. That process would involve occupying the floor continuously and engaging with Republicans in a procedural war of attrition, if they want to prevent the legislation from becoming law.

“It is literally a pressure cooker, and the ball is 100 percent in their court,” Representative Anna Paulina Luna, Republican of Florida and a leading proponent of the legislation, said of G.O.P. senators. “I think at this point in time, even Thune realizes this is kind of a political hand grenade if they don’t act.”

In a recent social media post, Mr. Trump insisted the bill had to pass “one way or the other,” even if it meant resorting to “a Talking Filibuster, à la ‘Mr. Smith Goes to Washington.’”