Sen. Charles Schumer (D-N.Y.) on Sunday predicted the Senate would pass its immigration reform bill by July 4 and said a strong vote for the measure could force House Republicans to embrace the Gang of Eight’s bill, despite Speaker John Boehner’s (R-Ohio) vow to the contrary.

Schumer said he and the other authors of the proposal hoped to win 70 votes, including nearly a majority of Republicans.

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The House is drafting its own series of immigration bills, and Boehner has said the lower chamber would not be “steamrolled” by Democrats. In a joint statement last month, the entire House Republican leadership said the House would not simply “take up and pass” the Senate bill.

Schumer still has his doubts.

“Congressman Boehner is in a box,” he said on NBC’s “Meet the Press.” “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 with the Senate bill,” Schumer continued. “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 thinking in the House among mainstream Republicans. And they may want to go for our bill.”

The full Senate is expected to begin debating the Gang of Eight measure on June 10.

In the House, the chairman of the House Judiciary Committee, Rep. Bob Goodlatte (R-Va.), told The Hill in an interview that the panel plans to vote on multiple immigration bills that address the issue in a piecemeal approach, rather than the comprehensive way envisioned by the Senate bill.

Goodlatte is still waiting for a bipartisan group of House negotiators to complete its own comprehensive legislation, which the members hope to unveil this month.

Appearing on “Fox News Sunday,” Goodlatte reiterated his preferred route on immigration.

“We think it’s better to do it by a step-by-step approach,” he said. At the same time, he said he would not rule out that the final measure would be a single piece of legislation.

Schumer also argued that the scandals dogging the Obama administration in the last several weeks would not interfere with the push for immigration reform, noting that three of the senators most critical of the administration on Benghazi and the IRS were members of the Gang of Eight and remained committed to the bill.

“These so-called scandals have not diverted us one iota,” Schumer said.