The Sun says Mourinho struck a deal with Inter president Massimo Moratti at the weekend and will take over from Roberto Mancini in the summer.
The Italian champions have not made an official announcement.
But Mourinho has told close pals he is going back into management and has begun drawing up a list of targets.
Midfielder Lampard, 29, enjoys a close relationship with Mourinho after they worked together for three years. Drogba has purchased an apartment in the centre of Milan and admits that Mourinho's departure rattled him earlier this season.