Stoke want a replacement lined up before they will consider letting the Wales international leave, having already turned down an opening £7m bid and explored a swap deal involving Hull forward Liam Millar without reaching agreement.
Thomas, who only joined Stoke from Huddersfield Town last year, has become a key figure under Mark Robins, contributing 10 goals and 13 assists in 47 appearances during his debut campaign at the Bet365 Stadium.
Hull City's interest reflects their determination to strengthen at the top end of the pitch after a perfect start to life back in the Premier League, having beaten Coventry and Manchester United in their opening two matches.
Thomas is understood to be keen on the move himself, with the chance to test his ability in the top flight proving a significant draw, but Stoke's stance has remained consistent throughout the saga: they will not sanction a sale until they know who would come in to replace him, with the Championship club having endured a difficult start to their own campaign.
Thomas made his name in non-league football at Boreham Wood before joining Huddersfield, where he spent five years, including loan spells at Blackburn Rovers and French side Nantes, and made 125 appearances before his move to Stoke.
He remains under contract at the Bet365 Stadium until 2028, giving Stoke no financial pressure to sell below their valuation, even with Hull's latest offer edging closer to the figure the Championship club are believed to want.
Sergej Jakirović's side have already shown a willingness to spend heavily to reinforce a squad adjusting to top-flight demands, and a successful pursuit of Thomas would add further attacking depth to a team that has made an eye-catching start to the season. With little time left before Tuesday's deadline, both clubs face pressure to resolve the situation quickly.