According to Sky Sports, Button becomes Brighton's fifth summer signing, after joining for an undisclosed fee.
Manager Chris Hughton said: "We're delighted to bring a goalkeeper of David's experience to the club."
"With Christian Walton and Robert Sanchez going out on loan this season, we felt we needed someone to provide further competition for places in that area of the team."
"He has a wealth of experience, having made over 300 appearances during his career so far, and I'm sure he will work well with Maty Ryan and Jason Steele."
The goalkeeper began his professional career at Tottenham, but went on a number of loan seasons, including to Shrewsbury and Plymouth.
He played for the England U20 team and went on to join Charlton in 2012, before making the switch to Brentford.
He's made over 60 appearances for Fulham, including 20 this season which saw Fulham gain promotion to the Premier League.