“In a way now, it’s like Straus and the right wing of the party don’t even speak the same language,” said Brandon Rottinghaus, the author of a new book, “Inside Texas Politics,” and a professor of political science at the University of Houston. “Straus is somebody who is still trying to hold onto the center of American politics and Texas politics. That’s becoming increasingly difficult to do.”

Mr. Straus shows no outward signs of feeling the pressure. In a recent interview in his office on the second floor of the Capitol behind the empty House chamber, Mr. Straus sat on a sofa with a tall glass of ice water and calmly paused for several seconds when asked whether Texas had shifted further to the far right since 2009.

“Well, a lot of the politicians have,” Mr. Straus replied. “I don’t think Texas has. And as the Republican Party continues to dominate all the statewide offices, the competition has been focused, really, on the small-turnout primaries.”

He added, “I’m not embarrassed to say that I know how to govern without being an extremist.”

Mr. Straus said he was not worried about the threats to vote him out after five terms as speaker. “If they want to try to change leadership, it’s in the rules how to do it,” he said. “I’ve been elected five times, so I have to know and be connected with where the members really are.”

The marquee item on the special-session agenda is the bathroom bill. But it is one of 20 pieces of legislation that will be debated, including property-tax reform, teacher pay raises and a ban on abortion coverage by private insurance plans.

The intraparty intrigue heated up a notch on Monday when Mr. Abbott told a conservative policy forum that he planned to publicize a list of lawmakers who support his 20-item agenda and those who oppose it, a list likely to have more political ramifications for Republicans than for Democrats.