Gov. Greg Abbott of Texas is engaged in a tense face-off with three of his state's largest cities over how local police officers work with federal immigration agents.

The governor gave the leaders of Houston, Dallas and Austin until Wednesday to amend their policies to his liking, or face losing more than $150 million total in public safety funding, including millions dedicated to providing security at World Cup matches this summer.

Republican state leaders have frequently tried to control the policies of their state’s Democratic cities, often through legislation, and sometimes through lawsuits. But in using critical funding as leverage, some consider this an echo of the Trump administration’s aggressive tactics.

“It’s a play out of President Trump’s playbook,” said Alejandra Salinas, a Houston councilwoman. “He thinks he can bully the city of Houston in the same way.”

Legal experts said the issue at hand — whether officers on the street can hold a person longer than usually permitted if the person is wanted by Immigration and Customs Enforcement — was not settled, and that Mr. Abbott could be testing the bounds of what support the police can provide federal immigration agents.

The fight began in Houston, where this month the City Council passed a new ordinance clarifying when its officers could detain people wanted by immigration agents. Mr. Abbott expanded the fight to Dallas and Austin after he said both had policies that also broke a contract with the state requiring the police to cooperate with immigration agents in exchange for funding.