A look at the politics behind Trump and Biden’s dueling border stops.

Republicans have long had an edge politically on the issue, with the G.O.P. advantage swelling even larger of late. In the fall of 2020, Mr. Trump was more trusted on immigration by a sizable 16 percentage points, according to NBC News polling at the time. That margin has more than doubled to 35 percentage points as of this January — the largest advantage either Mr. Biden or Mr. Trump had on any of the nine issues tested.

But Biden allies believe the recent decision by Republican congressional leaders — at Mr. Trump’s urging — to abandon a potential bipartisan border deal has provided the party a rare opening to cut into that deficit. The package would have made asylum claims more difficult, expanded detention capacity, increased fentanyl screening and paid for more border officers.

Democrats hope they can draw attention to the package’s failure and contrast Mr. Biden’s pursuit of bipartisanship with Mr. Trump’s belligerence.

“Donald Trump doesn’t want a solution,” said Gov. J.B. Pritzker of Illinois, a top Biden surrogate, in a call arranged by the Biden campaign before the Texas trip. “He wants a campaign slogan.”

Mr. Biden himself has said he would like to enact stricter policies enabled by the legislation. “If that bill were the law today, I’d shut down the border right now,” he said last month. Congressional Republicans have called on Mr. Biden to do so through executive powers.

The more immediate goal for Democrats is for Republicans to shoulder at least some of the blame for a border situation that even Mr. Biden himself described as “in chaos” at a fund-raiser in California last week. Border crossings set a monthly record of nearly 250,000 in December.

Mr. Biden and Mr. Trump are visiting Texas border towns hundreds of miles apart. Mr. Biden is headed to the Brownsville area and Mr. Trump will be in Eagle Pass, where Greg Abbott, the Republican governor, has clashed with federal officials over border security measures. Mr. Trump will later join the Fox News host Sean Hannity in Eagle Pass for a program that will run on Thursday night.

Mr. Trump had announced his trip first and Mr. Biden followed, though the president said on Monday he was unaware his predecessor would be there. “What I didn’t know is my good friend, apparently, is going,” he said.

Mr. Biden is not expected to announce any new actions on Thursday but instead to direct blame at congressional Republicans — and Mr. Trump — for the failed border package.

Representative Dan Crenshaw, Republican of Texas, said in an interview that Democrats were badly mistaken “if they think this one deal falling apart is somehow going to change the entire paradigm of voting on immigration.”

“Two things can be true at once: Republicans need to get their” stuff together, he said, using a word more colorful than stuff, “and change the laws that need to be changed. And it’s also true that the foundation of the problem is Biden’s policies.”

Mr. Crenshaw pointed to the more lenient actions and stance Mr. Biden took when he assumed office, including rolling back some Trump executive orders. “It just feels opportunistic unless he makes a big announcement,” he said of a Biden border visit.

For Mr. Trump, the flow of migrants across the border has been an animating issue from his first day as a presidential candidate in 2015, when he accused Mexico of sending drugs, criminals and rapists to America across the border.

More recently, Mr. Trump has zeroed in on a 26-year-old immigrant from Venezuela who was arrested in Georgia in the killing of Laken Riley, a nursing student. She was found dead on a wooded trail, and her death has turned into the latest flashpoint over crime and immigration.

The suspect, whom Mr. Trump has called a “savage monster,” had previously been arrested by Border Patrol for illegally crossing in September 2022 and was released with temporary permission to stay. He was arrested again in New York City in August. Mr. Trump has used the case to renew his call for the “largest deportation operation” in history.

Mr. Trump’s rhetoric on immigration has been especially dark. He has said those who are entering the country illegally are “poisoning the blood” of the nation. The phrase drew backlash, including for its echoes of language used by 20th-century autocrats, but it was well-received by Republican primary voters, with a CBS poll in January showing 82 percent of Republicans agreeing with the idea.

Interviews with voters in early primary states and public polling show that immigration is a deeply important issue to Republicans. In South Carolina, it rated as the top issue for 36 percent of voters — even higher than the economy. It was a close second in Iowa and New Hampshire, according to exit and entrance polling.

After the 2022 midterms, Democratic strategists who worked on House races said that internal research showed it was important for Democrats to directly address areas of vulnerability — at the time, that was inflation and the economy — in order to show voters that they empathize. Now, they said, that same approach should apply to immigration.

The recent special election for a House seat near New York City was a test case, as internal Republican polling showed 45 percent of voters rated immigration as the top issue — and the Democrat still won. There were plenty of extenuating circumstances, however. The reason a seat was open at all came from the historic expulsion of a scandal-plagued Republican former congressman, George Santos.

Representative Veronica Escobar, a Democrat who represents El Paso and is also a national co-chair of the Biden re-election campaign, said in an interview that it was clear immigration would be “absolutely a very potent political issue” in 2024.

“I hope that the American public sees that it is Republicans who refused to find solutions,” she said.

The stance of Ms. Escobar, who is a member of the Congressional Progressive Caucus, highlights the high-wire act Mr. Biden must walk on the topic to not alienate the Democratic base. Ms. Escobar herself was opposed to the border package, believing that it gave away too much to Republicans.