The Senate leader said his amendments will ensure that the new NSA system works.
Senators are expected to debate the USA Freedom Act after agreeing to move forward on the legislation Sunday.
Provisions of the Patriot Act expired at midnight.
McConnell's move means the first procedural vote could come as early as Tuesday.
"People here in town think I'm making a huge mistake," he said.
Paul voted against moving forward with NSA reform legislation.
"The senator from Kentucky needs to learn the rules of the Senate," McCain said
Reid blames McConnell, not Paul for the NSA stalemate.
Reid said that he was "saddened beyond words" to hear of Beau Biden's death.
The House will likely also consider trade legislation in June.