He wrote Boehner, a Catholic, that immigration is "a challenge that has confounded our nation for years, with little action from our federally elected officials. It is a matter of great moral urgency that cannot wait any longer for action."
The House has just a few legislative days left in the year, and prospects for any legislation are murky.
Responding to Dolan's letter, Michael Steel, a spokesman for Boehner, said the speaker "has been very clear that he supports common-sense, step-by-step reforms to fix our broken immigration system."
Most House Republicans reject a comprehensive approach as well as the Senate bill, with many questioning the offer of citizenship to people who broke U.S. immigration laws to be in this country. The House Judiciary Committee has moved forward with individual, single-issue immigration bills.
Although House Republican leaders say they want to resolve the issue, which has become a political drag for the party, many rank-and-file House Republicans have shown little inclination to deal with it. The bitter standoff with President Barack Obama on the budget and near default further angered House Republicans, who have resisted any move that might give Obama an immigration overhaul, the top item on his second-term domestic agenda.
Numerous House Republicans also are wary of passing any immigration legislation that would set up a conference with the Senate, fearing that they would lose out in final negotiations.
The Senate bill, strongly backed by the White House, includes billions for border security, a reworked legal immigration system to allow tens of thousands of high- and low-skilled workers into the country and a 13-year path to citizenship for those living here illegally.