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The bills were introduced in response to Secure Communities, a federal program that has become a cornerstone of the Obama administration's immigration enforcement strategy.

Under the program, the fingerprints of a suspect booked at a local jail are sent to the Homeland Security Department and compared with those in its files. If officials find that a suspect is in the country illegally or is a noncitizen with a criminal record, they may issue "a detainer" — a request that the police hold the person so that he or she can be transferred to federal custody.

The program has drawn the opposition of immigrants' advocates, and some elected officials across the country have sought to limit their jurisdictions' participation, in part by restricting cooperation with detainer requests.

The Council, which is dominated by Democrats, approved each bill by 40 to 7. Three Democrats joined the Council's four Republicans in opposition.

Mayor Michael R. Bloomberg is expected to sign both bills.

The bills ban the city from honoring detainers except in limited circumstances. For example, immigrants must have had a previous felony conviction or be facing a felony charge; or they must have been included in a federal gang database or on a terrorist watch list; or they must have had outstanding criminal warrants or previous orders of deportation; or they must have been charged with one of several misdemeanors, including sexual abuse, assault and gun possession.

In addition, the proposed laws would allow the city to honor a detainer request for an immigrant convicted of a misdemeanor as long as the conviction occurred within the past decade and was not for unlicensed driving, prostitution or loitering for the purposes of prostitution.

But the laws would block detainers for immigrants facing all but the most serious misdemeanor charges.

The measures expand the restrictions on detainers that were passed by the Council and signed into law by Mr. Bloomberg in 2011.

"We have seen too many families torn apart by current detention and deportation practices," the Council speaker, Christine C. Quinn, said in a statement. "Our legislation will ensure that the city does not enable such a harmful policy."

Still, the practical effects of the bills are unclear because they roughly echo guidelines issued recently by the director of Immigration and Customs Enforcement, or ICE, which handles detentions and deportations.

In a Dec. 21 memorandum, the director, John Morton, instructed field agents to issue detainers only for certain immigrants, including those who had been convicted of serious crimes, had been deported once and returned illegally or posed a threat to public safety.

The guidance, the agency said in a news release at the time, "restricts the use of detainers against individuals arrested for minor misdemeanor offenses such as traffic offenses and other petty crimes, helping to ensure that available resources are focused on apprehending felons, repeat offenders and other ICE priorities."