The 22-year-old, who made three appearances on loan at Bristol City in the first half of the season, has recently made his Albion comeback after a lengthy absence in which he faced a court case.
Dunk was eventually cleared of voyeurism and sexual assault last year but the trial meant that he was not involved as Brighton secured a top-six finish.
The defender has started Brighton's last two Championship games, keeping clean sheets in 2-0 wins against QPR and Bolton respectively, and assistant manager Nathan Jones was full of praise for him.
"I remember Gus picking him out from a young age and rated him and played him. Gus had a lot of faith in him, we have as well," he told the Argus.
"Lewis has had certain issues in terms of a year which he probably missed mentally. Now we've asked him, we've spoke to him, we are looking at him for a long-term future here.
"We see him as a Brighton player. We want him to be here for a long time.
"We want him to realise it isn't just ability but how you work, how you prepare, how you look after yourself, how you are off the pitch.
"That is all part and parcel of being a top Brighton player. He is starting to learn that now. He trains superbly well and that is proving dividends now in games."