The Brazil international, 19, opted to gain first-team experience away from Stamford Bridge after missing out on a place in Mauricio Pochettino’s squad, where he faces competition from expensive new signings Moises Caicedo and Romeo Lavia.
Santos will take part in Forest training on Friday with a view to making his Premier League debut against Manchester United on Saturday, a factor which was key to his decision to sign.
He told the Forest club website: “I am very happy and excited to be here. I know that Forest is a big club, they play good football and they have good players, which is important for me.
“It has always been my dream to play in the Premier League and I can't wait to enjoy playing.”
Chelsea are hoping to trim their squad further before next week’s transfer deadline, with Romelu Lukaku, Trevoh Chalobah, Callum Hudson-Odoi and Conor Gallagher all thought to be available for bids.
Ross Wilson, Forest chief football officer, added: “Everyone knows how talented Andrey is, so naturally we are pleased that amongst the considerable interest in him, he and Chelsea have chosen Nottingham Forest. We are all delighted to have him.”