It has been reported by Sky Sports that Chelsea manager Frank Lampard has been impressed with the impact that Silva has made at the club so far this season.
The Premier League outfit are reportedly keen on taking up the option to keep the Brazil international central defender next season as well.
The 36-year-old joined Chelsea on a one-year contract in the summer of 2020 on a free transfer after leaving French giants Paris Saint-Germain at the end of last season.
It has now been reported that the Blues plan to trigger the 12-month extension option, although official discussions with the former AC Milan star will not begin until the New Year.
Silva has scored one goal in five Premier League matches and has played thrice in the Champions League for Chelsea so far this season.
The London club are third in the Premier League table at the moment with 18 points from nine matches, just two points behind leaders Tottenham Hotspur and second-placed Liverpool.
Chelsea have also progressed to the last-16 stage of the Champions League from Group E.