Hull have now upped their offer and are now inching closer to a full agreement to sign the 25-year-old.
The Ghana international emerged as a top target for the Championship side as they look to bolster their attacking options after losing star winger Jaden Philogene and striker Liam Delap.
Thomas-Asante has impressed for West Brom since joining from Salford in 2022 and has finished as top scorer in his first two seasons at The Hawthorns.
The Saints are looking for a replacement for Che Adams after he departed St Mary’s at the end of his contract to join Torino.
But Hull now have the edge in the race after successful negotiations with the Baggies.
Thomas-Asante has less than 12 months to run on his deal with the Midlands side and they have now agreed to cash in and avoid losing him for free in 2025.
He notched 12 goals and two assists in 43 appearances across all competitions in the 2023-24 campaign, starting in 33 of those outings.
Since joining West Brom from Salford in 2022 he has finished as top scorer in his first two seasons, proving his ability in front of goal.
At the other end of the field, Hull have allowed James Furlong to join AFC Wimbledon on a season-long loan.
The Tigers will kickstart the new campaign on 10 August at home against Bristol City, as they look to push on after finishing just outside the play-offs last season.
