The 24-year-old has agreed a four-year deal and becomes the Whites’ fifth summer signing and could make his Leeds debut in Sunday’s Championship opener against Stoke City.
Bamford said: “I’ve been needing to play up front and wanting to be the main man somewhere for a long time. Leeds have given me that opportunity and it’s something I’ve grabbed with both hands.
“I know some big players have worn (the No 9 shirt) and there probably is a bit of pressure, but I’ll try and do my part and even when I don’t get goals I’ll always chip in for the team.”
Bamford began his career at Nottingham Forest and made two substitute appearances before being signed by Chelsea aged 18 for £1.5m in January 2012.
He failed to make a first-team appearance at Chelsea and had loan spells at MK Dons, Derby, Boro, Crystal Palace, Norwich and Burnley.
Middlesbrough paid £6m to seal a permanent move for him in January 2017 and he scored 13 goals in all competitions last season, including a hat-trick against Leeds at the Riverside in March.
Leeds signed Jack Harrison on loan from Manchester City on Monday having added defender Barry Douglas from Wolves for an undisclosed fee over the weekend. Chelsea pair Lewis Baker and goalkeeper Jamal Blackman both arrived on season-long loan deals earlier this month.
Liverpool midfielder Ben Woodburn, 18, is close to agreeing a season-long loan move to Championship side Sheffield United.
The Blades look to have beaten off competition from Aston Villa for the teenager, who had a medical at Bramall Lane on Tuesday.
Sheffield Wednesday chairman Dejphon Chansiri will hold a fans’ forum on Monday evening.
The Owls are yet to make a signing this summer amid unconfirmed reports of a transfer embargo.
A club statement said: “Mr Chansiri will invite questions from supporters on any subject relative to SWFC and pledges complete openness and transparency.”
Doncaster Rovers defender Tyler Garratt is poised to join AFC Wimbledon on loan.