The Blues signed Silva on a one-year deal but have the option of extending his contract by a further 12 months.
According to Tuttomercatoweb Chelsea are already looking to exercise that clause following his impressive start to the 2020-2021 season.
Silva has brought defensive solidity to a side that was notorious for its shaky back line.
Having featured nine times across all competitions so far, Silva has achieved six wins and three draws.
Not only has Silva improved Chelsea’s fortunes defensively, he has provided better passing statistics than many of his team-mates – successfully completing a pass 95.3 per cent of the time.
Chelsea have also increased their average goals per game to 2.6 since Silva joined the club.
Therefore, Lampard is keen to extend Silva’s time in London.
It is believed negotiations could be accelerated this week as Silva has already declared his interest to extend his contract.
He said: ‘I am very proud to be coached by someone who has been one of the best midfielders in the world and to have faced him as a player.
‘We exchange our ideas in the locker room and that’s a very good thing. He talks to me a lot and I try to help my best, on and off the pitch, in order to keep playing in the best possible way.
‘I have learned in [a lot of different competitions] and I use it to help young players.
‘This is my first experience in the Premier League. I hope to continue to play well and that my contract will be renewed.
‘I will do my best possible and really contribute to this team, this new project. I’ve accepted this new challenge, it’s going to be really good for me. ‘
While Chelsea director Marina Granovskaia previously shared her vision for Silva at the club.
Claiming he will help the Blues win trophies and compliment their exciting young talents.
She said: ‘We are delighted to be able to add a player of Thiago Silva’s proven world-class credentials to our squad.
‘Having performed at the highest level for many years, we have no doubt his experience and quality will complement the many exciting talents we already have here.
‘Thiago will be an excellent fit and we hope he can add some new trophies to his impressive list of honours.