He said Rep. Robert Goodlatte, the Republican chairman of the House Judiciary Committee, is working on "a set of principles to help guide us as we deal with this issue."
The Senate bill, passed in June, would provide a path to citizenship for the 11 million immigrants living in the country illegally and tighten border security.
The House Judiciary Committee has approved piecemeal bills, but they have languished since the summer despite intense pressure from religious groups, business, labor and immigration advocates.
Although House Republican leaders say they want to resolve the issue, which has become a political drag for their party, many rank-and-file Republicans have shown little inclination to deal with immigration.
Many House Republicans are wary of passing any immigration legislation that would set up a conference with the Democratic-controlled Senate, fearing the House could lose out in final negotiations.
Democrats expressed frustration over the House's lack of action at a House Judiciary Committee hearing on enforcing laws to deal with legal immigrants who overstay their visas.
"If we can vote 45 times to gut Obamacare and have another vote scheduled this week, why can the Republican leadership not find the time to schedule one vote on immigration?" asked Rep. Luis Gutierrez, a Democrat. "They say that where there is a will there is a way and I think there is time to get something done, but I don't believe the Republican Party has the will to do it."