Specter said President Bush has agreed to sign legislation that would authorize the secretive Foreign Intelligence Surveillance Court to review the constitutionality of the National Security Agency's most high-profile monitoring operations.
"You have here a recognition by the president that he does not have a blank check," the Pennsylvania Republican told his committee.
Attorney General Alberto Gonzales said the administration supports Specter's bill.
"My understanding from the president is that the legislation could be very helpful," Gonzales told reporters. "It would continue to allow the president to gather up information to protect the country."
Since shortly after Sept. 11, 2001, the NSA has been eavesdropping on the international calls and e-mails of people inside the United States when terrorism is suspected. Breaking with historic norms, the president authorized the actions without a court warrant.
The disclosure of the program in December sparked outrage among Democrats and civil liberties advocates who said Bush overstepped his authority as president.
Specter said the legislation, which has not yet been made public, was the result of "tortuous" negotiations with the White House since June.
"If the bill is not changed, the president will submit the Terrorist Surveillance Program to the Foreign Intelligence Surveillance Court," Specter said. "That is the president's commitment."
If there are any changes to the bill before it's passed, they must be to the president's satisfaction or he won't go along, CBS News correspondent Sharyl Attkisson reports.
Specter said the court would make a one-time review of the program rather than performing ongoing oversight of it.
An administration official who spoke on condition of anonymity said the bill's language gives the president the option of submitting the program to the intelligence court, rather than making the review a requirement.
Specter thinks his newly crafted bill – the details weren't finished up until 10 p.m. Wednesday night – will pass his committee, the full Senate and probably the House, too, but there are no guarantees, Attkisson reports.
But there are serious jurisdictional issues between the Intelligence and Judiciary committees which still need to be ironed out, reports CBS News correspondent Gloria Borger.