WASHINGTON (CNN) -- Senate Majority Whip Dick Durbin, part of the bipartisan Gang of Eight senators pushing immigration reform, said Sunday on CNN's "State of the Union" that he agrees with Republican Sen. Marco Rubio that their bill may need some improvement but argued that they need to "stand by the basic agreement" of the legislation.
The Democrat from Illinois suggested, however, that he would support a same-sex provision in immigration reform, a plan that could undermine bipartisan support for the bill introduced last month.
Florida's Rubio, the leading Republican on immigration reform, said last week that the bill he helped craft would struggle in the Senate and would not pass in the House as it currently stands. The bulk of public outcry has been over the bill's promise for a pathway to citizenship for the 11 million to 12 million undocumented immigrants in the country.
"Congress will have a chance to make it tougher, yet still realistic," he wrote in an opinion piece for the Wall Street Journal, responding to further criticism that the bill is not strong enough in border security.
Durbin said there may be "elements where we need some improvement" but did not specify which parts should be tweaked and argued instead that the main foundation of the bill should stand.
"We've got to basically stick to the standard of what we've established, what we've agreed over the last three months," he told CNN's chief political correspondent Candy Crowley.
In the proposed legislation, it would take 10 years for undocumented immigrants to get green cards and another three years to gain citizenship. Along the way, they would have to pay a fine and back taxes and pass a background check. The process would also be contingent on bolstering security.
The legislation mandates strict new controls on the U.S.-Mexico border before any pathway to legality is opened.
The bill goes for markup this week in the Senate Judiciary Committee.
The committee's chairman, Democratic Sen. Patrick Leahy of Vermont, is proposing legislation that would allow gay Americans to sponsor their partners not born in the U.S. for green cards.
Currently, such rights apply to only to foreign-born men or women in heterosexual marriages with U.S. citizens. If Leahy moves ahead, some fear that it could upend efforts for any significant Republican support for the compromise immigration effort.
Durbin, a co-sponsor of the Leahy legislation, described the measure as a "hot" and "contentious" issue but added that he nonetheless supports it.
"If we can find a way through this to protect that basic right of an individual and still pass immigration reform, that's what I want to achieve," he said. "We have two very important issues before us here. I hope we can get them both done. We may face a choice at some point in the future."
President Barack Obama indicated at a news conference in Costa Rica on Friday that the idea was "the right thing to do" but acknowledged that such a politically charged issue may ultimately not make it into a final immigration package.
"I'm not going to get everything I want in this bill," he said in response to a question about his support for gay rights and its relation to immigration reform, a priority for Democrats seeking to maintain strong support from Hispanics and Republicans trying to woo them to their side.-CNN's Dana Davidsen, Alan Silverleib, and Kevin Liptak contributed to this report.
From our media partner CNNAshley Killough