WASHINGTON — House Republicans on Tuesday staked out what they cast as a middle-ground option in the debate over immigration, pushing an approach that could include legal residency but not a path to citizenship — as their Democratic counterparts favor — for the 11 million illegal immigrants already in the country.

Republicans also signaled that they are open to the idea of breaking immigration legislation into several smaller bills, which would allow them to deal with the question of highly skilled workers, as well as a farmworker program, without addressing what Democrats and immigration advocates say is the larger issue of potential citizenship. Immigration advocates favor a comprehensive measure to enable them to use elements that have bipartisan backing to build support for broader legislation.

At a House Judiciary Committee hearing exploring an overhaul of the immigration system — the first of several such hearings expected in the House — Representative Robert W. Goodlatte, Republican of Virginia and chairman of the committee, tried to frame what he called the question of the day: “Are there options that we should consider between the extremes of mass deportation and a pathway to citizenship for those not lawfully present in the United States?”

It was a question later echoed by Representative Lamar Smith, Republican of Texas and the former chairman of the committee, when questioning Mayor Julián Castro of San Antonio. “Do you see any compromise area between the current status quo and a path to citizenship for virtually all the 11 million who are illegal immigrants in the country today?” he asked.