The Brazilian defender joined Chelsea on a free transfer from PSG last year and has appeared 32 times this season.
Thiago Silva is set to lose his contract within a month, but transfer expert Fabrizio Romano says the 36-year-old is determined to keep working.
Both teams are said to be happy with the contract extension, although no official confirmation is expected ahead of the UEFA Champions League final.
Chelsea are also believed to be in preliminary talks with midfielders Jorginho and N’Golo Kante about extending their stay at Stamford Bridge.
