Some of President Trump’s staunchest allies on Capitol Hill have told him in no uncertain terms that they won’t back his threat to “close” the U.S.-Mexico border, describing the move as economic suicide.

In particular, both of Texas’s GOP senators have told him not to do it.

Sen. Ted Cruz said he sympathizes with Mr. Trump’s frustration at the soaring number of illegal immigrants who’ve figured out how to game the U.S. immigration system, with children and families surging into the country in record numbers.

“But the answer is not to punish those who are legally crossing the border,” he said. “The answer is not to punish Texas farmers and ranchers and manufacturers and small businesses. Closing legal points of entry would harm American commerce and legal transit between Mexico and the United States, and leave coyotes and human traffickers to roam free in the wilderness of our unsecured border.”

Sen. John Cornyn warned Mr. Trump directly in a phone call Tuesday, according to the Dallas Morning News.

“I know you are very frustrated,” Mr. Cornyn told Mr. Trump. “Let us try to work on some other approaches.”

But Mr. Cornyn told the newspaper he wasn’t sure his message got through.

It’s not clear what other approaches might work for getting Congress to tackle the growing border crisis.

Key House Democrats revealed their own plans this week for dealing with the 100,000 illegal immigrants a month now jumping the border, and none of them involved changing the “pull factor” incentives Mr. Trump says are encouraging people to make the trip.

Instead, Democrats said the solutions lie in Central America, where they said the U.S. should pump more taxpayer money and work on cooperative agreements.

Their solution for the U.S. side of the equation was a better deployment of personnel to be more welcoming to those who do arrive at the border illegally.

That approach has Mr. Trump frustrated. He’s already signaled he is pulling money from Central America, not putting more in.

“Democrats are leaving us absolutely no choice,” White House press secretary Sarah Huckabee Sanders told reporters Wednesday. “Eventually, it may be that it’s the best decision that we close the border.”

• Dave Boyer contributed to this article.

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