Sen. Robert Menendez questioned Speaker John A. Boehner’s commitment to an immigration overhaul after the Ohio Republican said Tuesday he could not see bringing a bill to the floor that did not have majority Republican support.

“It is amazing and alarming that Speaker Boehner would allow a minority of House members — who will never, ever support immigration reform — to dictate the fate of bipartisan, comprehensive reform that an overwhelming majority of the American people want,” said the New Jersey Democrat. Menendez is a member of the bipartisan Senate “gang of eight” that negotiated the chamber’s immigration overhaul. “You have to question the Republican leader’s seriousness about real immigration reform if he is willing to put tea party politics ahead of the will of the American people. That’s not in our national interest and it’s certainly not about fixing our broken immigration system.”

“No matter what he has said, there’s going to be significant national pressure on the House to do something on immigration,” said Senate Majority Leader Harry Reid, D-Nev., in comments similar to those made earlier Tuesday by House Minority Whip Steny H. Hoyer, D-Md. “I’m only worried about what’s going to happen here.”

Tennessee GOP Sen. Lamar Alexander said likewise, pushing for the bill to get through the Senate by Independence Day. “My own view is the best thing for senators to do is to do our job — and then send the thing over to the House and let them do their job. So, I don’t worry about it, and I don’t think most of us do,” Alexander told reporters. “We’ve been presented with an opportunity to end de facto amnesty, to create a legal immigration system, and secure the border and if we can do that, we should do that it — and we should do it in the next two weeks.”

Alexander added that a border security agreement would be key to his own vote for the package.

Some Democrats are also pushing amendments to strengthen immigration controls. For instance, Sen. Joe Manchin III of West Virginia wants to impose a requirement that requires individuals seeking expedited citizenship under the DREAM Act serve in the military or actually graduate from college. “It doesn’t eliminate anybody from being on a pathway to citizenship,” Manchin said of his amendment. “But, if they’re going to fast track it, they should complete something. If they go to the service, they have to have a completion of … duty.”

Sen. Claire McCaskill, D-Mo., wants amendments to enhance internal enforcement.

“Whether it’s tracking overstays on visas or whether it’s going after employers who repeatedly violate the law. I’ve got an amendment that would put more teeth into repeated violations under e-verify,” McCaskill said. She added that it would be “wildly inappropriate” to make demands for specific amendments before voting for the bill, however. Asked if she had any concerns that Reid could find himself in the position of filing cloture on the immigration bill before her amendments get considered, McCaskill was realistic.

“If somebody tells you they’re not afraid of that, they don’t — clearly [they] have only been here 10 minutes,” she said.