"We're going to put immigration on the floor starting on June 10. I predict it will pass the Senate by July 4," Sen. Chuck Schumer, D-N.Y., a member of the bipartisan Senate group often referred to as the "Gang of Eight," which drafted the immigration-overhaul legislation, told the NBC News program "Meet the Press."
"We're hoping to get 70 votes, up to 70 votes, which means a lot of Republicans," he said Sunday. "And we're willing to entertain amendments that don't damage the core principles of the bill but improve the bill, just as we did in committee."
Schumer said he hoped House Republicans would be swayed to follow the Senate if the measure gets the strong Senate Republican support he predicted, even though an aide to House Speaker John Boehner, R-Ohio, said last week Democratic senators were wrong to think House Republicans would rubber-stamp the Senate bill just because it got strong Senate GOP support.
"Boehner is in a box," Schumer said. "There are about 60 or 70 of his people who are against any immigration reform.
"But at the same time, he knows that the Republican Party will be consigned to a minority party for a generation if they're anti-immigration.
"So my advice to him is, let's see what happens in the Senate bill. If we can come out of the Senate with close to a majority of the Republican senators and almost every Democrat, that may change the equation in the House and the thinking in the House among mainstream Republicans. And they may want to go for our bill."
The proposed Border Security, Economic Opportunity and Immigration Modernization Act -- which includes a path to citizenship for the 11 million immigrants in the United States illegally -- passed the Senate Judiciary Committee 13-5 last month, with three Republicans joining the Democratic majority.
House Judiciary Committee Chairman Bob Goodlatte, R-Va., told "Fox News Sunday" he didn't expect the House would accept the Senate bill, at least not as a whole.
"We think it's better to do it with a step-by-step approach," he said. "We've introduced a series of bills in the House, both the Judiciary and the Homeland Security Committee. We'll continue down that path.
"Our legal immigration, our enforcement, and figuring out the appropriate legal status for people who are not lawfully present in the United States all need to be addressed.
"The final outcome ... in terms of the form of the [House] legislation is not yet known."