The senator said the agency had agreed to help locate Democratic state legislators who departed to try to block a vote on congressional maps. Democrats said federal law enforcement was being misused.

Senator John Cornyn, Republican of Texas, said on Thursday that the F.B.I. had agreed to his request to help track down dozens of Democratic Texas state lawmakers who left the state to prevent a vote on a redistricting plan.

The activation of federal agents would create a standoff between the Trump administration and Democratic state leaders in Illinois, where many of the absent Democrats have taken refuge.

But by midday on Thursday, local law enforcement officials in Texas had not made any requests to the F.B.I. for assistance, and there were no indications that the Justice Department or the bureau had yet gone to court to seek warrants to track the lawmakers, according to a person familiar with the situation.

Nor was the bureau expecting to take immediate action, although that could always change, the person said. In text messages, Kash Patel, the director of the F.B.I., told Mr. Cornyn that the bureau would offer its assistance, according to two people briefed on the exchange.

“The reality is that all that he has said is that the F.B.I. has been authorized to locate the House Democrats — nothing more,” Gov. JB Pritzker of Illinois, a Democrat, told reporters at the Illinois State Fair on Thursday, accusing Mr. Cornyn of “a lot of grandstanding.”

“I welcome the F.B.I. coming to the state,” he added. “But they won’t be arresting anyone because there is no federal law that prohibits those Texas House Democrats from being here.”

Mr. Cornyn’s statement may well have been intended as an intimidation tactic, some Democrats said, but it was also issued with a political backdrop. Locked in a tough Republican primary fight, Mr. Cornyn and his opponent, Texas’ hard-right attorney general, Ken Paxton, have been competing for days to look tough with the runaway Democrats.

Several of the Texas Democrats who are in Illinois said that as of Thursday afternoon, no federal agents had been seen or reported at the lawmakers’ hotel in St. Charles, about 25 miles west of Chicago.

The whereabouts of the Texas lawmakers are widely known, but until now at least, they had considered themselves safe from arrest because they were far from the jurisdiction of Texas law enforcement agencies.

“I am proud to announce that Director Kash Patel has approved my request for the F.B.I. to assist state and local law enforcement in locating runaway Texas House Democrats,” Mr. Cornyn said in a statement.

Some of the Democratic lawmakers said they received a security briefing on Thursday afternoon, though they declined to discuss its details or what agency had provided it. Their hotel was evacuated because of a bomb threat on Wednesday morning.

Others expressed concern about increasing pressure and tension.

“I’m terrified that the rhetoric coming from both my colleagues, the governor, the attorney general and the president is putting a dangerous target on our backs,” said Representative Mary González, an El Paso Democrat. “Words have consequences, and in a moment in time like right now, when you’re saying, ‘Hunt them down. Here’s our location. Go get them,’ what do they think is going to happen?”

Still, the Texas Democrats also reiterated their determination to remain outside the state.

“I will not be intimidated into silence by politicians who are trying to rig our democracy to cling to power,” State Representative Ann Johnson, a Houston Democrat, said in a statement.

In his letter on Tuesday requesting the bureau’s assistance, Mr. Cornyn argued that the Democratic lawmakers might have violated state bribery laws by accepting money from outside groups that support their efforts to prevent a vote on a redrawn political map.

No criminal warrants have actually been filed against the lawmakers. The speaker of the Texas House has issued civil warrants, saying that the Democrats violated rules of attendance for that chamber, but those warrants have not been considered enforceable outside the state during previous legislative walkouts.

Mr. Cornyn has not suggested that federal statutes have been violated. Instead, he said the F.B.I. could help state law enforcement when scofflaws leave the state.

The “F.B.I. has tools to aid state law enforcement when parties cross state lines, including to avoid testifying or fleeing a scene of a crime,” he wrote. “Specifically, I am concerned that legislators who solicited or accepted funds to aid in their efforts to avoid their legislative duties may be guilty of bribery or other public corruption offenses.”

For Mr. Cornyn, the effort to force the Texas Democrats to return to Austin has figured prominently in his re-election campaign, which first must clear a primary against Mr. Paxton, a popular figure with Republican voters in the state.

Mr. Paxton has said that if Democratic lawmakers do not return in time for a roll call in the Texas House at 1 p.m. Friday, he would petition the Texas Supreme Court to have some of their seats declared abandoned and therefore vacant. Gov. Greg Abbott has filed a lawsuit on the same lines against the leader of the walkout, Representative Gene Wu of Houston.

However, Mr. Paxton said in a podcast interview this week that such a process was untested and would be a “challenge.”

As an indication of how the U.S. Senate race and the fight over the legislative walkout have intertwined, Mr. Cornyn’s campaign took Mr. Paxton’s statements and rapidly turned them into an attack ad, claiming that the attorney general was not being aggressive enough against the Democratic lawmakers.

Mr. Paxton, for his part, said his office was investigating an organization run by former U.S. Representative Beto O’Rourke for helping to fund the walkout, and accused Mr. O’Rourke of being “scared of accountability.”

“Scared?” Mr. O’Rourke responded on social media on Thursday. “If we stop the Texas steal & win the House, your guy’s crimes & corruption will be exposed for all to see,” he added, referring to President Trump.

Democratic leaders in Washington attacked the involvement of the F.B.I. as a misuse of federal law enforcement by the Trump administration. “Shouldn’t the F.B.I. be tracking down terrorists, drug traffickers and child predators?” Representative Hakeem Jeffries, the House minority leader, said in a statement. “We will not be intimidated.”

Dozens of Democratic Texas state lawmakers have been in Illinois since Sunday, when they left Texas to deny Republicans the quorum needed to hold a vote in the Texas House on the proposed congressional map. They were joined by other Texas lawmakers, who initially traveled to New York to meet with Gov. Kathy Hochul and discuss possible redistricting by Democrats there.

The proposed Texas map, requested by Mr. Trump, would remake five districts currently held by Democrats so that they would favor Republican candidates in the 2026 midterm elections, when Democrats nationwide are expected to gain seats.

The redistricting fight in Texas has rapidly ballooned beyond the state’s borders. Democratic-led states like Illinois and California have been threatening to redraw their own political maps in response to Texas, and Republicans are looking at other states they control, including Missouri and Indiana, to follow Texas’ lead.

Glenn Thrush contributed reporting from Washington.

J. David Goodman is the Houston bureau chief for The Times, reporting on Texas and Oklahoma.

Julie Bosman is the Chicago bureau chief for The Times, writing and reporting stories from around the Midwest.