(CNN) — (CNN) -- Two top lawmakers in the House immigration reform effort battled Sunday over how a compromise could be obtained in the Republican-controlled chamber, after the Democrat-controlled Senate passed sweeping reform legislation last week.
Republican Rep. Bob Goodlatte, chairman of the House Judiciary Committee, flatly said the House should not take up the Senate bill and should produce its own version instead. But Democratic Rep. Luis Gutierrez, a fellow member of the committee, said Republicans are failing to produce anything worth voting on. The two sat side by side Sunday morning on CNN's "State of the Union."
"We cannot put compromise to one side," Gutierrez said. "What the House Republicans are doing is giving a Republican solution, and a Republican solution isn't what we saw was successful in the Senate."
In a 68-32 vote, the Senate passed a 1,200-page bill Thursday that promises to overhaul immigration laws. The legislation includes a pathway to citizenship for most of the country's 11 million undocumented workers. A last-minute, bipartisan amendment added tougher border security measures to the bill as a way to make it more enticing to House Republicans.
A bipartisan group is also working on a package in the House, though it differs sharply from the Senate measure by making it harder for undocumented immigrants to get on a path to citizenship.
Members of the House group negotiating the bill would also require that border security measures be in place before any process toward gaining legal status could begin. Lawmakers agreed to include security "triggers" in their proposal in hopes of attracting support from more House Republicans who have been highly critical of the Senate bill. The Senate rejected a similar GOP proposal.
Pointing to the majority of Republicans that voted against the bill in the Senate, Goodlatte said he wants to "work with Democrats" to produce a bill, "but not the Senate bill."
"The compromise is going to have to come both in getting a bill out of the House and then in going to conference with the Senate to work out the differences," he told CNN's chief political correspondent Candy Crowley.
Asked if he thinks the House bill will include a pathway to legalization, he said "yes," but not necessarily a pathway to citizenship.
Goodlatte reiterated what House Speaker John Boehner has said, saying the bipartisan group must produce a bill "that can get a majority of House Republicans to support it."
But Gutierrez said the proposals coming from Republicans are "old" and "discredited."
"The breadth and the depth of support which exists for comprehensive immigration reform is greater than I've seen on any other issue," he said, talking about labor unions, religious communities and poll numbers that show support for immigration reform.
But he added House Republicans are "the one people that stand on the side, unwilling to compromise and sit down with Democrats."
President Barack Obama said Saturday he called Boehner and House Democratic Leader Nancy Pelosi to encourage action on the immigration front.
"Now the ball is in the House's court," he said at a joint press conference with the president of South Africa. "I've called both speakers - Speaker Boehner and Leader Pelosi - and encouraged them to find a path to get this done. And the framework that the Senate has set up is a sound framework."
"Our diversity is a source of strength. So we need to get this right. We can be a nation of laws and a nation of immigrants," he added, urging the House to get a bill passed by the August recess. "There's more than enough time. This thing has been debated amply, and they've got a bunch of weeks to get it done. And now is the time."
Democratic Sen. Chuck Schumer, a member of the Senate Gang of Eight, said on "Fox News Sunday" that Republicans fear primary challenges if they support immigration reform, but he still predicted the House would pass the Senate bill by the end of this year.
Republican Sen. John McCain, also a member of the Gang of Eight, said on the same program that Congress ultimately responds to public opinion and will "get in line behind everybody else."
"I really hesitate to tell Speaker Boehner exactly how he should do this. But I think Republicans realize the implications of the future of the Republican Party in America if we don't get this issue behind us," he said.
According to a CNN/ORC International Poll released earlier this month, a bare majority of Americans supported the Senate bill being considered at the time. That poll came before the amendment that toughened the bill's border security. Sixty-two percent said the main focus of the country's immigration policy should be border security, while 36% said it should be a pathway to citizenship.