His offer came during a 10-minute telephone discussion and was prompted by the violence at the World Cup qualifier at the weekend.
The two leaders agreed that the violence at the match in Rome should not cause any diplomatic row between the two countries, but also that there needed to be a proper investigation into what happened.
Downing Street said it was unable to say whether Mr Blair would take up the invitation.
Mr Prodi's telephone call came as the last of the 31 fans who were arrested following the match were released.
An embassy spokesman said that some of them had been given consular help to get hotel accommodation and flights back to England.Reuse content