President Barack Obama may find that he has some surprising new Republican allies next year.

With Democrats poised to lose seats and possibly control of Congress in the midterm elections, House Republicans are looking to the next year to restart their relationship with the White House. Members say they think Obama will have incentive to work with an emboldened GOP and see an opportunity to advance a shared agenda.

Rep. Jeff Flake (R-Ariz.), who introduced a broadly supported immigration reform bill with Rep. Luis Gutierrez (D-Ill.) in 2007, said the issue has a shot in the 112th Congress if Republicans grow their House ranks. The issue remains a priority for Obama, who has faced pressure to act on it since failing to deliver on a promise to pass it during his first year in office.

“I think it’s easier for Republicans to do it,” Flake said. “I mean, we’ve seen that the Democrats didn’t even come close. There wasn’t even a real effort. Republicans have that, at least.”

He conceded that Congress is “a more toxic environment” now than it was in 2007. But the problem is that the only legislation put forward in the past year was “something the unions backed” and lacked a temporary worker plan, a must for comprehensive reform, Flake said.

“You’ve got to do it in an off-year, and it’s still a long shot,” he said. “But it’s more likely under Republicans than Democrats.”

Republicans are also predicting movement on several free-trade agreements that have stalled. Obama has said he supports moving forward on pending Bush-era trade agreements for Panama, Colombia and South Korea.

“I’m convinced that, if the president will bring those free-trade agreements to the House, to Congress, with his full support, we can pass all three of them,” said Rep. Kevin Brady, ranking member of the Ways and Means Subcommittee on Trade.

To date, Obama has been “saying all the right things” on trade but hasn’t acted because of the “politics” of appealing to his Democratic base, namely labor unions, the Texas Republican said.

But that will change if Republicans gain seats or take control of the House, Brady said, adding that he already has “a great working relationship” with U.S. Trade Representative Ron Kirk, a fellow Texan.

“He can do a tremendous job if the president gives him the green light,” Brady said.

A Senate GOP leadership aide said Democrats who are criticizing trade deals may be in for a surprise next Congress. “They’re running around demonizing trade right now, but the president has said he wants some of these trade agreements,” the aide said.

On another front, Education and Labor ranking member John Kline said he has already been working “really closely” with Chairman George Miller (D-Calif.) and Education Secretary Arne Duncan on reauthorization of the No Child Left Behind law, the standards-based education reform put into place by President George W. Bush.

The Minnesota Republican said that win or lose on Nov. 2, his party will have to work with Democrats and the administration to reauthorize the 2001 program because there is widespread consensus that it “has to be dealt with.”

“I think we’re going to be able to work on it together,” Kline said. “Not because we agree on a lot of things, but because we agree that we need to fix No Child Left Behind. I think it’s got to be a bipartisan effort. It’s the only way we can fix it.”

Michael Steel, a spokesman for Minority Leader John Boehner (Ohio), said Republican leaders welcome the idea of working with Obama on jobs and spending.

“If Washington Democrats are finally ready to address those issues, we look forward to working with them,” he said.

But not every Republican sees 2011 as a year of collaboration with the White House.

For all Obama’s talk about finding a bipartisan solution to bring down the deficit, the White House has yet to reach out to the GOP for ideas in this arena, Budget ranking member Paul Ryan said.

“They don’t talk to us at all,” the Wisconsin Republican said.

And conservative Rep. Steve King said he could not think of any areas of common ground with the president.

“He’s a committed ideologue. I think it’s going to be very hard for him to move to the right as Bill Clinton did and triangulate,” the Iowa Republican said.

Several senior Democratic aides predicted gridlock will characterize the 112th Congress.

“Ha!” a senior Senate Democratic aide responded when asked about a shared GOP-White House agenda. “Republicans will oppose everything, just as they have the last two years.”

Another senior Democratic aide suggested that the GOP would have far fewer moderates than it did in 1995 and won’t want to cut deals with a president seeking re-election.

Republicans will “have an eye toward 2012,” this aide said. In addition, tea-party-inspired Republicans will be beholden to “extreme fundamentalist politics that doesn’t deliver results,” the aide said.

Still, some aides pointed to smaller-ticket issues that could move forward. For one, Republicans could work with the White House on the highway reauthorization bill, a typically bipartisan, earmark-laden measure. But even that could run afoul of a GOP majority on a mission to rein in government largesse.

Aides also pointed to issues relating to deficit reduction, if the president’s deficit commission comes back with recommendations, and energy legislation, so long as it doesn’t include cap-and-trade language.

But Flake dismissed the suggestion that an energy bill is among the items that might advance next year.

“There’s no appetite on the Democratic side to bring that up,” he said. “And if we’re in control, certainly nothing is going to move.”

Steven T. Dennis and John Stanton contributed to this report.