The 23-year-old is in advanced talks with the club where he enjoyed a successful loan spell in the 2014-15 season.
Bamford enjoyed a fine season with Boro two years ago, scoring 19 goals and winning the Championship Player of the Year award.
However, in loan spells at Crystal Palace, Norwich and most recently Burnley, Bamford has struggled for game-time and form, and has failed to score since the end of the 2014-15 season.
The deal is said to be worth an initial fee of £6m, with £4m in add-ons, meaning the transfer will have a potential value of £10m.
Therefore, Bamford could net Chelsea a £8.5m profit, despite failing to play a single game for the Blues, having signed the striker for £1.5m from Nottingham Forest back in 2012.