The Seagulls have acquired Webster, 24, on a four-year deal in light of Manchester United seemingly close to announcing Harry Maguire’s arrival from Leicester City for £80m.
And Dunk has been linked with the Foxes as Maguire’s replacement, meaning Webster’s arrival could see the England international move before next week's deadline.
“Adam is a player the club has been aware of for sometime, and someone we came up against last season at Swansea City,” said new Brighton boss Graham Potter, who joined the south-coast club from Welsh side Swansea in May.
“He is a quality defender, comfortable on the ball, fits the profile of the type of player we want to bring to the club, and he was very keen to join us.
“He is still relatively young, but has good experience from the number of games he has played. Predominantly a centre-half, he can also play at right-back, and he will bring additional competition alongside our existing defensive options.”
Webster played 44 Championship games for City last season, starting in all but one of his appearances and scoring three goals.
The move betters the £17m paid for Iran forward Alireza Jahanbakhsh last year.
Brighton beat off relegation last year, finishing 17th and have since swapped managers, hiring Potter to replace Chris Hughton.
They begin the new campaign against Watford on August 10.