The C.I.A. is paying AT&T more than $10 million a year to assist with overseas counterterrorism investigations by exploiting the company’s vast database of phone records, which includes Americans’ international calls, according to government officials. The cooperation is conducted under a voluntary contract, not under subpoenas or court orders compelling the company to participate, according to the officials. The C.I.A. supplies phone numbers of overseas terrorism suspects, and AT&T searches its database and provides records of calls that may help identify foreign associates, the officials said. The company has a huge archive of data on phone calls, both foreign and domestic, that were handled by its network equipment, not just those of its own customers.