What - if anything - should Obama do about immigration?

President Obama vowed again in an interview that aired Sunday on "Face the Nation" to press ahead with executive actions reforming America's immigration system, saying House Republicans had over a year to pass the Senate's bipartisan reform bill, and they failed to act.

He added that Congress still has time to pass a bill, saying it would take time to implement any executive actions, and that legislation would supersede any administrative changes he might make.

But with a triumphal midterm election behind them and control of the Senate ahead, Republicans continued warning the president that acting unilaterally on immigration would be a huge mistake.

"I believe it would hurt cooperation on every issue," said Sen. John Barrasso, R-Wyoming, a member of the Senate Republican leadership, on "Fox News Sunday." "It would be like the president pulling the pin out of the hand grenade and throwing in as we're trying to actually work together."

Barrasso and other top congressional leaders attended a meeting on Thursday at the White House, where Republicans pressed Mr. Obama to give Congress time and space to find a legislative solution on immigration.

"I'm hoping that cooler heads at the White House can prevail upon the president to say, 'Look, if you want to have a good, constructive final two years of your presidency, don't do this now, wait until the new Congress is sworn in, let them come together,'" Barrasso said Sunday.

Rep. Cory Gardner, R-Colorado, who won his race for the U.S. Senate on Tuesday, told "Fox News Sunday" he hopes the president "will change his mind" and allow Congress to work its will on immigration.

"If he does this, then I'm concerned that he won't be doing the right thing," Gardner said, "and that would hurt our ability to move forward the next two years."

Rep. Darrell Issa, R-California, told ABC News that Mr. Obama risks spoiling his opportunity to sign a bill if he moves on his own.

"I'm hoping the president will delay and have a real comprehensive discussion about what's possible because a great deal is possible on immigration reform," Issa said.

And former House Majority Leader Eric Cantor, R-Virginia, whose loss to a primary challenger this cycle was blamed, in part, on a willingness to entertain immigration reform proposals, said the president would be in for a "very, very difficult" few years if he flouts Congress on the issue.

"If he does that, that's incendiary," Cantor told NBC News.

Democrats, though, said Republicans had their opportunity, and they blew it, pointing to the failure of the Republican-controlled House to take up the bipartisan Senate bill or advance a proposal of their own.

"I think the president has been very patient," said Rep. Xavier Becerra, D-California, who was also at Thursday's meeting at the White House. "He's been waiting a year and a half for the House Republicans to act on the bill that the Senate passed on a bipartisan basis."

"House Republicans continue to insist that the president must wait to help fix what everyone agrees is a broken immigration system," he added on "Fox News Sunday. "So, I think the president is right to move forward."

Rep. Pete Gallego, D-Texas, told CNN that Democrats and immigration reform advocates have been "very frustrated" by Republicans' inability to act on the issue since the Senate passed its bill last June.

"In the meantime, there's a lot of our families are getting deported," he said. "So, as I've said even before winning this seat, we do think there should be executive action."

At least one Democrat, though, seemed willing to entertain the argument that the president should wait and see what the new Congress can produce.

Sen. Chris Coons, D-Delaware, told CNN the president should delay his action "only if there's a clear commitment from the Republican leadership to take up debate and adopt an immigration reform bill."

"They've had more than a year almost a year and a half now since the Senate pass a broad bipartisan bill that was led in part by Republican senators and have taken no action on it," Coons said.