Ronny Deila stressed his desire to keep Fraser Forster at Celtic following the 4-0 victory over KR Reykjavik.
Executive vice-chairman Ed Woodward insists Manchester United are ready and willing to break the British transfer record.
The 23-year-old, who has been in fine form for Brighton this season, will remain with the club until the summer of 2017.
Manager Gus Poyet told the club's official website: "We are absolutely delighted to have agreed this new deal with Will. He has been our match-winner many times, and has been one of our key players since he arrived.
"He has earned this new deal with his performances on the pitch. I'm pleased he is settled and so happy here. It is great for us that he has committed his long-term future to the club."