House Speaker John Boehner (R-Ohio) answers questions during a press conference at the U.S. Capitol on Thursday. (Win McNamee/Getty Images)

House Speaker John A. Boehner (R-Ohio) said Thursday that immigration reform is “absolutely not” a dead issue, but he offered no timetable for when he might schedule a vote on legislation.

“I’m hopeful we can make progress on this very important issue,” Boehner said at his weekly news conference on Capitol Hill.

House Republicans have shown little appetite for pursuing a comprehensive overhaul of border-control laws after the Senate approved a broad bipartisan plan in June. Some GOP leaders have said there is not enough time to deal with immigration before the end of the year, alarming advocates who fear that passing a bill could be even more difficult with the 2014 midterm elections approaching.

But Boehner insisted Thursday that his colleagues are still pursuing smaller-scale immigration bills, and said he was encouraged by recent suggestions that President Obama could support a piecemeal approach. At a forum with business executives this week, Obama said the House could divide a comprehensive plan into smaller measures, provided that each of the bills was approved.

Apparently alluding to the president’s health-care law, Boehner said “the American people are skeptical of big, comprehensive bills, and frankly, they should be. The only way to make sure immigration reform works this time is to address these complicated issues one step at a time. I think doing so will give the American people confidence that we’re dealing with these issues in a thoughtful way and a deliberative way.”

But Minority Leader Nancy Pelosi (D-Calif.) expressed skepticism in an e-mail to reporters: “The American people are waiting, Mr. Speaker. The time is now for a vote on immigration reform.”

Even if the House were to move forward in a piecemeal fashion, the sticking point remains a path to citizenship for the nation’s 12 million illegal immigrants. The Senate bill would allow most of those immigrants to pursue citizenship over a 13-year period, a measure that most House Republicans oppose.

House committees have focused instead on bills to tighten border controls and toughen laws governing undocumented immigrants.

Advocates have continued to keep the pressure on. A coalition of Republican governors supporting immigration published a full-page advertisement in USA Today on Thursday, and activists with the America’s Voice pro-immigration group delivered Thanksgiving turkeys, processed by immigrants, to Boehner’s office.

“Immigrants are working hard for America, why can’t he?” the group said in a message urging constituents to call the speaker in support of immigration.

In his news conference, Boehner reiterated that he thinks “Congress needs to deal with this issue. Our committees are continuing to do their work. There are a lot of private conversations that are underway to try to figure out: How do we best move on a common-sense, step-by-step basis to address this very important issue?”