The Dutch defender ran out of contract at the end of last season but could stay with the club.
He was included in a list of retained players published by the Premier League.
It is understood talks between Veltman and the Seagulls are ongoing with nothing concluded as yet.
Solly March, Adam Webster and James Milner have parted company with the club in the last three weeks, although Milner was offered a new deal.
Veltman is hugely popular with players, coaches and fans and would offer experience and right-back cover in a busy season which will include Conference League action.
He has played 181 times for the club since arriving for less than £1 million
But he saw compatriot Mats Wieffer move ahead of him in an adaptation of the right-back role last season.
Albion are widely expected to bring in Portuguese right-back Costinha from Olympiacos when the transfer window opens next week.
Versatile defender Igor Julio looks set to leave the club with a return to his native Brazil possible.
Meanwhile under-21s right-back Charlie Tasker is being lined up for a loan move when he recovers from a thigh injury.
The official Premier League released list has been announced online, and alongside the likes of Solly March and Adam Webster.
However, it appears to be a new contract on the horizon for club hero and right-back Joel Veltman! According to Charlie Parker-Turner.
Most fans were expecting a similar departure for Veltman, but it looks like the tide has certainly changed.
Marking nearly six years at the club, Veltman has gone down as one of the best signings and players in the modern Albion era. Since the promotion, very few have come close to him for his reliability, consistency, and value for money.
Veltman was bought for just £900,000 from Ajax in 2020 and rapidly became a fan’s favourite. Despite many Ajax fans being relatively unsure of him and even questioning his move to the Premier League, Albion fans will always have a different opinion
