The central midfielder had been a member of the Reds' 25-man squad for the Premier League Asia Trophy, but the Reds have since given Stewart permission to discuss terms and undergo a medical with Hull.
The Championship outfit's offer includes add-ons which could take the current fee believed to be around £5.5 million up to an eventual £8 million.
Reds to make profit on midfielder they signed for free
Even the initial fee represents a sizeable profit on the 23-year-old, who only joined the club on a free transfer under Brendan Rodgers in the summer of 2014.
Stewart moved to Merseyside upon the expiry of his contract at Tottenham Hotspur and made 20 first-team appearances for the club after his debut in an FA Cup clash with Exeter City back in January 2016.
Stewart had only signed a new five-year contract at Anfield back in November after Liverpool had rejected interest from a handful of clubs including Swansea City, West Bromwich Albion and Newcastle United back in January.
But Stewart made only two appearances, both in the FA Cup, in the second-half of the campaign and Hull's offer is one considered too good to turn down for a bit-part player.
Midfield is also one of Liverpool's strongest areas, with Stewart unlikely to have featured over Jordan Henderson and Emre Can. Liverpool are also keen on RB Leipzig's Naby Keïta, who excelled at No.6 last term though he can also play as a box-to-box midfielder.
That would leave Stewart's chances limited and they will allow the Englishman, who has spent time on loan with Cheltenham Town, Burton Albion and Swindon Town since joining Liverpool, to move on.
Robertson and Stewart set to switch clubs with Kent another to possibly follow
Liverpool are set to seal the signing of Hull left-back Andrew Robertson in the coming days with the Scotland international due to undergo a medical on Merseyside on Thursday.
The two clubs agreed an £8 million up-front fee with a further £2 million in add-ons for the 23-year-old, who made 33 Premier League appearances last term.
Another deal between the club could see Ryan Kent join Hull in a season-long loan move. The 20-year-old winger, a member of the squad in Hong Kong, is attracting interest from Leonid Slutsky after a fruitful year with Barnsley in the second tier last term where he won the Tykes' Young Player of the Season award.
Birmingham City are among the other Championship clubs keen on Kent, though the ongoing deals between Hull and Liverpool could help them beat off competition to sign the England youth international.