Cruz suggested that Reid hoped to move the bill quickly to avoid scrutiny from the American public.
“I think he's starting to get nervous because the American people are starting to pay attention," he said.
Backers believe they will have the 60-plus votes needed to pass the legislation. But many immigration reform proponents are hoping for as many as 70 votes, believing that an overwhelming majority could pressure the reluctant GOP House to take up the bill.
Many Republican senators say they want tougher border security measures before endorsing the bill.
Cruz said the bill’s backers “think they can get 70 votes in the Senate and they can use that to bludgeon the House into adopting this legalization first and never secure the border approach.”
"There are probably 20 Republican senators in the U.S. Senate who are on the fence, who are wobbling on this issue and not sure of how to vote," Cruz added.
Top House lawmakers, though, say the Senate proposal is dead on arrival, and that they will focus instead on crafting their own legislation.