Rubio: Current Senate immigration bill can't pass the House

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“That is a very legitimate suspicion — it’s one that I share — and if there’s anything we can do to make it even tighter, to ensure that the laws are enforced, so that we don't have this problem in the future, that’s exactly what we should be working on," Rubio said.

Still, the Florida lawmaker and possible 2016 presidential candidate said the Senate bill should serve as a jumping-off point for lawmakers in the House.

"I think this is a starting point that obviously we can and should improve," Rubio said.

Rubio also defended the legislation as a superior alternative to the status quo, noting that the government currently had no way to track those who overstayed their visas and that parts of the border remained unsecured.

"If we don't do anything we're stuck in what we have in place now," Rubio said.

At a press conference Tuesday at the White House, President Obama said he would be "open-minded" about an alternative proposal from House Republicans.

"The bottom line, though, is, is that they've still gotta meet those basic criteria: Is it making the border safer?" Obama said. "Is it dealing with employers and how they work with — with the governments to make sure that people are not being taken advantage of or taking advantage of the system? Are we improving our legal immigration system? And are we creating a pathway for citizenship for the 11 million or so who are undocumented in this country?"

The president added that he had "been impressed by the work that was done by the Gang of Eight in the Senate," even though the bill they produced was "not the bill I would have written."