A lawmaker who helped negotiate a bipartisan bill to overhaul immigration predicted on Sunday that comprehensive legislation would overwhelmingly pass the Senate by July 4 while House Republicans cautioned that they would write their own version, one piece at a time.

Sen. Chuck Schumer, D-N.Y., said he anticipates as many as 70 of the 100 senators will vote for the measure heading to the full Senate on June 10. Even if it passes there, the proposal faces tough prospects in the Republican-led House, where lawmakers are at work on their own piecemeal approach that could stall a pathway to citizenship for the estimated 11 million immigrants living in this country illegally.

“We are moving forward because we believe in a bipartisan way this is so vital for America, and we’ll have a good bill,” Schumer said, pledging to allow colleagues to amend the legislation.

Not so fast, House lawmakers cautioned.

“That Senate bill is not going to move in the House,” said Rep. Ileana Ros-Lehtinen, R-Fla.

For months, four Democratic senators met with four Republican colleagues behind closed doors and developed a proposal that would enact new border controls and enforcement mechanisms in the workplace, allow tens of thousands of workers into the country legally for high- and low-skilled jobs and create a 13-year path to citizenship for those already living here illegally. It passed through the Senate Judiciary Committee last month by a vote of 13 to 5; three Republicans joined the Democratic majority.

House lawmakers, though, have pledged to put together their own measure — likely taking components of the comprehensive Senate plan one at a time and adding their own priorities.

“We think it’s better to do it with a step-by-step approach,” said House Judiciary Committee Chairman Bob Goodlatte, R-Va.

“We’ll continue down that path, but the final outcome in terms of the form of the legislation is not yet known,” he added.

Democrats and Republicans alike recognized the political potency of the issue. The Senate, led by Democrats, is putting added pressure on the House, led by Speaker John Boehner, R-Ohio.

“Congressman Boehner is in a box. 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,” Schumer said.

In 2012, President Barack Obama won re-election with 71 percent of Hispanic voters and 73 percent of Asian voters backing him. A thwarted immigration overhaul could send those voting blocs more solidly to Democrats’ side.

“We are hard at work on this problem. We have a broken immigration system in the country, it needs to be fixed, 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,” Goodlatte said.

But it is unlikely to be a sweeping answer in the House, lawmakers said.

“I don’t know if we’ll have comprehensive reform or we will have it piece-by-piece. But that Senate bill may not even pass the Senate itself,” Ros-Lehtinen said.

Schumer spoke to NBC’s “Meet the Press.” Ros-Lehtinen was interviewed on CNN’s “State of the Union.” Goodlatte was on “Fox News Sunday.”

(AP)