Lampard was appointed earlier this month as Maurizio Sarris replacement and is understood to have impressed Chelseas players with his intense training sessions during his early stages at Stamford Bridge.
Mourinho, meanwhile, remains out of work after he was sacked by Manchester United last December.
Lampard was one of Mourinhos key players at Chelsea and the pair won two Premier League titles as well as the FA Cup during their time together at Stamford Bridge.
But despite their success, Mourinho declined the opportunity to send a message of support to Lampard during his trip to Silverstone on Saturday.
Asked by Sky Sports if Lampard is the right appointment for Chelsea, Mourinho waved his finger and replied: I dont speak about football. My football? I have no football now.
Asked if he is hungry to return to management, Mourinho replied: I am but I will wait for the right club, for the right project.