Responding to critics who claim that the secret services have "cleaned up" the documents and that many key files are missing, Basescu said he did not rule out that possibility.
"I can't guarantee that ... no files were stolen before 2005," he said.
The file of Ion Iliescu, the former president, could not be found by a panel set up by Basescu earlier this year to investigate communist regime's crimes.
Iliescu has denied ordering the secret services to destroy his file.
Romania's Securitate spied on the country's 23 million citizens, and had an estimated 700,000 informers, some as young as nine, when Nicolae Ceausescu was overthrown in December 1989.
Some contents of the files have been made public in recent months, revealing that politicians, journalists and other public figures collaborated with the Securitate.
After the secret police was disbanded, the files were held by the Romanian intelligence service, the Romanian foreign intelligence service and the defence ministry.
On Thursday, the three institutions said that the handover had been completed.
The files are now under the control of the national council for study of the Securitate archives, although 75,000 files relating to counter espionage and terrorism remain in the archives of the Romanian intelligence service.
Members of the council have access to these files, which remain secret because they are considered still relevant to national security.