Senate Majority Leader Mitch McConnell said Tuesday that he would permit a vote on House-passed legislation to restrict the National Security Agency’s authority to review data about Americans’ phone calls.

“I certainly think we ought to allow a vote on the House-passed bill,” said McConnell, R-Ky. “If there aren’t enough votes to pass that, we need to look at an alternative.”

The House passed the bill, known as the USA Freedom Act, last week by a vote of 338-88, putting considerable pressure on McConnell to allow a vote on the Senate companion by Republican Mike Lee of Utah. It would reauthorize section 215 of the 2001 Patriot Act, the provision that undergirds the NSA program, but set new limits on it.

McConnell opposes the House bill because it would bar the NSA from continuing its current practice of collecting records of all Americans’ phone calls in its own database.