(CNN) –

Comprehensive immigration reform, approved last month by the Senate, still doesn’t have a clear path forward in the House of Representatives, lawmakers said Sunday.

Even the bill’s supporters, like Rep. Raul Grijalva, D-Arizona, weren’t optimistic that the measure that passed with bipartisan support in the upper chamber would find a way forward.

“They can pick it to death.” Grijalva said of Republican lawmakers, speaking on CNN’s “State of the Union.” “They can try to take away a pathway to citizenship, which are nonstarters for many of my colleagues. Or they can go forward and allow something comprehensive, something that is bipartisan and something that will move us off this position. But right now, the comments by (House Speaker John) Boehner have been anything but encouraging.”

Boehner has said since the spring that the House would not take up the Senate’s version of immigration reform, which includes a pathway to citizenship for undocumented immigrants that hinges upon new requirements for bolstering border security.

Even after Republicans huddled Thursday to determine a way forward on immigration reform, it was still unclear what measures the body would take up. GOP lawmakers reiterated their opposition to the Senate bill and said they would take their time in crafting legislation that fulfills their wishes for even greater strengthening of border security.

The question of a pathway to citizenship, however, still divides Republicans. Participants in the meeting described a 50-50 split over the undocumented immigrant issue.

While some Republicans have labeled any pathway as “amnesty,” others have been urging their party to embrace a track to legal status as a way to show the party’s commitment to fixing a broken system and addressing concerns of Hispanic Americans — the nation’s largest minority demographic, which voted heavily for Democrats in last year’s election.

That argument was blasted Sunday by Rep. Steve King, an Iowa Republican who’s remained firmly opposed to allowing undocumented immigrants a pathway to citizenship.

“It’s a mistake for Republicans to believe the election was about immigration,” King said on “Fox News Sunday.” “I don’t remember a debate between Barack Obama and Mitt Romney on immigration. I remember waking up the morning of the 7th of November, after the election, and hearing immigration was the reason that Mitt Romney wasn’t president-elect on that day.”

King expressed skepticism at the effectiveness of any immigration law, including one bolstering border security, saying he doubted President Barack Obama would effectively enforce the new provisions.

But even Republican Senate Leader Mitch McConnell said Sunday the House should vote on some kind of immigration measure, saying the current system was sorely in need of a revamp.

“I’m a big fan of what legal immigration has done for our country,” McConnell said, pointing to his wife, Elaine Chao, who was born in Taiwan and went on to become U.S. labor secretary under George W. Bush. “I hope, even though the Senate bill in my view is deficient on the issue of border security, I hope we can get an outcome for the country that improves the current situation. I don’t think anybody is satisfied with the status quo on immigration, and I hope the House will be able to move forward on something.”

McConnell appeared on NBC’s “Meet the Press,” as did Sen. Harry Reid, the Senate majority leader, who said the House would have to vote on immigration since so many Americans support some kind of change to the system.

“Yes, they will act,” Reid said. “They have to. This is something that the vast, vast majority of Republicans, Democrats, and independents support, and John Boehner should let the House vote.”