The governor of Texas, Greg Abbott, is telling big city Democratic mayors that he will be sending buses full of asylum-seeking migrants north until the Biden administration, as he puts it, “does its job by securing our border.”

In a letter to the mayor of Chicago, Lori Lightfoot, Mr. Abbott said he agrees with her assertion that the “situation is completely untenable.”

“This is not a Texas problem — this is a problem for the entire United States of America,” he said. “Until Biden secures the border to stop the inflow of mass migration, Texas will continue this necessary program.”

Mr. Abbott was responding to recent complaints from Ms. Lightfoot and other mayors about the Texas policy, which has led to thousands of migrants being shuttled to cities such as Washington, Chicago, New York, and Philadelphia from border cities in Texas. In a letter to Mr. Abbott, Ms. Lightfoot said 8,000 have arrived in Chicago alone in the last several months.

“We simply have no more shelters, spaces or resources to accommodate an increase in individuals at this level,” Ms. Lightfoot’s letter said. “I know by your actions that you either do not see or do not care about the trauma these migrants have already faced under the humanitarian crisis you created.”

New York’s mayor, Eric Adams, joined the fray Monday in defense of Ms. Lightfoot, suggesting that Mr. Abbott’s actions were motivated by racial animus. “It is impossible to ignore the fact that Abbott is now targeting five cities run by Black mayors,” Mr. Adams said. “Put plainly, Abbott is using this crisis to hurt Black-run cities.”

The charge, Mr. Abbott’s office responded, is ludicrous. The migrants all sign voluntary consent forms on which they agree on their destinations. “Mayor Adams is once again spreading falsehoods and outright lies. He knows full well these migrants willingly chose to go to New York City,” Mr. Abbott said.

The war of words between the GOP governor and Democratic mayors is likely to intensify in the coming weeks after the public health measures that have allowed authorities to turn away asylum-seekers at the southern border — known as Title 42 — end on May 11 with the expiration of the Covid public health emergency.

El Paso and other Texas border towns, which have borne the brunt of the influx, are bracing for a renewed surge as thousands of migrants from around the world who have been waiting in makeshift camps on the Mexican side of the border cross over and formally apply for asylum. El Paso declared a state of emergency Monday in anticipation of the influx.

The Biden administration announced Tuesday that it would be beefing up border control efforts in the coming days by deploying as many as 1,500 active-duty military troops to provide logistical support for officers with the U.S. Customs and Border Protection agency that normally control border access. The president’s press secretary, Karine Jean-Pierre, said the troops would not be involved in law enforcement or be dealing with immigrants directly.

Democrats in Congress and immigrants’ rights advocates widely panned President Trump when he deployed military assets to the border in 2018. Kevin Appleby, of the Center for Migration Studies, told the New York Times that the move was a sign of weakness instead of strength. “The president’s deployment of the world’s strongest military against a band of vulnerable asylum seekers is embarrassing,” Mr. Appleby said.

In his letter to the Chicago mayor, Mr. Abbott said there are estimates that as many as 13,000 immigrants will attempt to cross the border every day after the Title 42 protocols end. He cited reports that there are more than 35,000 waiting to cross into El Paso alone once the Title 42 expulsions end. “If Chicago can’t deal with 8,000 in less than a year, how are small Texas border communities supposed to manage 13,000 in just one day?” he asked.

To provide relief to El Paso and other cities, he pledged that the policy of busing migrants to northern “sanctuary cities” would continue, adding that more cities may soon be added to the list currently receiving the buses.