Inter pushed to sign Giroud in January, but ultimately couldn't agree terms with Chelsea, despite the veteran having less than six months to run on his contract.
Gazzetta dello Sport says Conte remains keen on a reunion with his former Blues striker.
Giroud's minders have tabled their terms to Inter GM Beppe Marotta - a two-year contract worth €5m. Marotta is now considering the proposal.
And as Marotta mulls over things, Conte is staying in regular contact with Giroud to assure the Frenchman he is wanted in Milan.