WASHINGTON (CNN) – Two days after Barack Obama called for the administration and Congress to investigate the breach of the presidential candidates’ passport data, the Senate Judiciary Committee’s senior Republican said he agreed with the Illinois senator.

“There are federal criminal statutes involved. I think that ought to be a very intense investigation. I think privacy is a very fundamental matter,” Pennsylvania Sen. Arlen Specter told Gloria Borger on CNN’s Late Edition Sunday. "I think it ought to be something for Attorney General Mukasey, and I think that it may well be something for the Senate Judiciary Committee, where I’m the ranking member."

Democratic Sen. Ron Wyden of Oregon agreed with Specter, adding that the Government Accountability Office had been warning about this problem for a decade, and that the White House deserved a share of the blame for fostering a “a culture of disregard for privacy.”

Thursday night, the State Department confirmed that Obama’s passport file had been improperly accessed three times this year. On Friday, they revealed that the passport files of the rest of the presidential field – Sens. Hillary Clinton and John McCain – had been similarly breached.

Secretary of State Condoleezza Rice on Friday publicly apologized for the actions of the three contract employees, and said the State Department inspector general would investigate the matter. State Department officials also briefed Senate staffers for the three candidates and for Senate Foreign Relations Committee chairman Joseph Biden of Delaware.

–CNN's Jessica Rummel