Sen. Charles Schumer, a member of a bipartisan group working on changes to U.S. immigration law, said the lawmakers are "on track" to reach an agreement by the end of this week.
The Senate immigration effort has centered on creating a 13-year path to U.S. citizenship for the estimated 11 million undocumented immigrants in the U.S., which Democratic President Barack Obama has made a priority for his second term.
"All of us have said that there will be no agreement until the eight of us agree on a big specific bill," Schumer said. "But hopefully we can get that done by the end of the week. That's what we're on track to do."
Sen. Lindsey Graham (R-S.C.), who is also part of the negotiating team, said on NBC's "Meet the Press" that he expected a bill in the "next couple weeks." A business-labor compromise was reached March 29 to resolve a dispute over a proposed guest-worker program that had been impeding Democratic-controlled Senate efforts to change immigration law.
The heads of the U.S. Chamber of Commerce, the nation's largest business lobbying group, and the AFL-CIO, the biggest labor federation, agreed to a tentative deal that would expand the number of low-skilled laborers that can come into the U.S. while restricting those allowed to work in construction, where unemployment is high.
'LOTS OF LITTLE ISSUES'
Graham said lawmakers have been "revisiting" that agreement.
"The Democratic Party has to give us a guest-worker program to help our economy," Graham said. "That's what we're arguing over."
McCain said the legislation should include a path to grant citizenship to people in the U.S. illegally, tougher border security and a guest-worker program to allow qualified non-U.S. citizens to work legally.
There isn't a single sticking point holding up a bill, Schumer said. "There are lots of little issues on different parts of the bill that we have to resolve," he said.
Both Schumer and McCain said passing a final bill remains difficult.
"There will be a great deal of unhappiness about this proposal because everybody didn't get what they wanted," McCain said. "There are entrenched positions on both sides of this issue."
"There's a long road," Schumer said.
Schumer said he also thinks it's possible that a gun control bill will advance.
"I'm still hopeful that what I call the sweet spot, background checks, can succeed," he said.
"I don't understand it," McCain said. "What are we afraid of?"
Republican senators including Marco Rubio of Florida said in a March 22 letter to Senate Majority Leader Harry Reid, a Nevada Democrat, that they "intend to oppose any legislation" that infringes on the right to bear arms or the ability of Americans to do so "without being subjected to government surveillance."