The announcement of Silva’s arrival appears imminent after the Brazilian underwent a medical in Milan.
The Telegraph claim Silva has penned a one-year deal with an option for a further year, but the centre-back will not become Chelsea’s new skipper.
Silva has limited English and would struggle to communicate with much of the squad and Lampard in his first months at Stamford Bridge.
Cesar Azpilicueta currently holds the armband and has been the club skipper since 2018.
Silva, 35, said his goodbyes to PSG fans on Wednesday, saying on Instagram: ‘Today a cycle ends, after eight years with the PSG team I would like to thank my team-mates, the whole technical team and management, the fans, my family, God and friends, for all these happy moments we experienced in the city of light.’
Silva will become Chelsea’s fifth major summer signing after the arrivals of Hakim Ziyech, Timo Werner, Malang Sarr and Ben Chilwell.
Chelsea have backed Lampard in the transfer market, spending big to land his top targets and the club are also closing in on deal for Kai Havertz.
The Bayer Leverkusen star is set to be the club’s biggest outlay of the summer with a £90million transfer free agreed.
More than four players have tested positive for Covid-19 less than two weeks before the start of the new Premier League season.
Chelsea kick off their season with an away trip to Brighton on 14 September.