The House is expected to vote this week on a bill that would bar the Justice Department from giving grants to cities that refuse to comply with federal immigration enforcement officials, a measure that Republicans hope can bridge the reopened chasm within the party over immigration as they try to avoid alienating Hispanic voters.

The Senate Judiciary Committee on Tuesday seized on renewed attention to the issue: Its members rushed to introduce their own proposals to crack down on noncompliant jurisdictions and heard testimony from relatives of victims of crimes committed by illegal immigrants.

Among those testifying was the father of Kathryn Steinle, whose murder this month has been highlighted by Donald J. Trump, a Republican presidential candidate, intensifying the debate. The suspect in her killing is an illegal immigrant with an extensive criminal record who was released by the authorities in San Francisco despite a deportation order pending against him.