House Republican leaders have decided members will vote this week on legislation that would deny millions of dollars in federal funding to "sanctuary cities" that do not enforce federal immigration laws.

The legislation was added Monday to the House schedule, and according to a spokesman from the office of the Majority Leader. The earliest that vote could come is Thursday, since the House Rules Committee will meet Wednesday evening to agree on rules for debate and voting on the legislation.

The "Enforce the law for Sanctuary Cities Act," would withhold several kinds of federal grants for police and immigration services in cities that intentionally ignore immigration laws.

The legislation is sponsored by Rep. Duncan Hunter, R-Calif., and while he first proposed the measure in 2011, it has become a sudden priority for the GOP leadership in the wake of the the July 1 shooting death of Kate Steinle on a San Francisco pier.

The shooter, Juan Francisco Lopez-Sanchez, is an illegal immigrant felon who had been deported five times. Police in San Francisco, a sanctuary city, did not turn him over to federal officials who were seeking to deport him, and did not inform federal officials when he was released.

The House measure could cost sanctuary cities tens of millions of dollars in policing grants and grants to help cities cope with the influx criminal illegal immigrants. Specifically, it would shut down grants that states use under a criminal alien assistance program, and cut grants under the Community-Oriented Policing Services program to states with policies that go against federal immigration law.

States that prohibit law enforcement from gathering information about citizenship or immigration status would also be denied funding.

The bill has wide support from House GOP lawmakers, and is one of several legislative responses that have come in the wake of Steinle's death.

Rep. Matt Salmon, R-Ariz., has introduced "Kate's law," which would mandate a five-year prison sentence for anyone who returns to the United States after being deported.

"By instituting mandatory minimums for those who illegally reenter our nation after already having been deported, we help dissuade those who so casually disregard our laws and continue to victimize Americans," Salmon said.