TV news channels reported that Gilani had been provided a three-room suite on the sixth floor of the President's official residence.
He had earlier moved into the Islamabad Club after he was disqualified by the Supreme Court.
The news channels quoted their sources as saying that Gilani could not stay at the club due to security concerns.
Some reports said Gilani was expected to play a behind-the-scenes role in guiding the administration of new Prime Minister Raja Pervez Ashraf.
Ashraf was on Friday elected the Prime Minister during a special session of the National Assembly following Gilani's disqualification.
Gilani was convicted by the apex court in April after he refused to reopen graft cases in Switzerland against President Asif Ali Zardari.