The 35-year-old became a free agent, following Paris Saint-Germain’s 1-0 defeat to Bayern Munich in the Champions League final on Sunday.
Silva spent eight years with PSG after joining from AC Milan in 2012.
He won seven Ligue 1 titles during his spell in France.
The Brazil defender will become the Blues’ third signing of the summer, following the arrivals of Hakim Ziyech and Timo Werner.
Chelsea could also complete deals for Leicester left-back Ben Chilwell and Bayer Leverkusen forward Kai Havertz in the coming days.