The Tigers are pursuing an £8million transfer for the 23-year-old, who has made 11 first team appearances at the Reds since signing from Tottenham Hotspur's academy in 2014.
The deal could be a make-way for Andrew Robertson to move in the opposite direction, with Liverpool keen to find a left-back this summer.
Stewart is highly thought of at Anfield and was given a run of games by Jurgen Klopp in cup competitions last season.
Leonid Slutsky's transformation of the Hull playing staff continued yesterday as he completed the capture of Crystal Palace striker Frazier Campbell on a two-year deal.
Meanwhile, Hull defender Ahmed Elmohamady is poised to reunite with former boss Steve Bruce at Aston Villa.
Sheffield United were boosted with four players sealing new contracts on Tuesday but their pursuit of Southend United's Ryan Leonard remains in the balance.
The League One champions are rivalling Millwall for the player but apparently had a £500,000 bid knocked back on Friday.
Southend chairman Ron Martin insists the player will not be sold and will instead be held to the last year of his contract.
Leeds United have completed the loan signing of talented Everton defender Matthew Pennington.
The centre-back, who scored in the Merseyside derby last season, is Thomas Christiansen's 10th signing of the summer.
Pennington, 22, will fly over to Austria to link up with his new team-mates on a pre-season tour.
Sheffield Wednesday have reportedly utilised their scouting network of the non leagues and taken a punt on Wealdstone defender Ciaron Brown.
With defensive options short, boss Carlos Carvalhal could be attempting to pull a rabbit out of the hat with the 20-year-old who impressed as the Stones finished eighth in Vanarama National League South last season.
Of the defender's talent, Wealdstone boss Gordan Bartlett said: “We have a great pedigree of bringing young players on and moving them up to the pro ranks and Ciaron is more than capable of making the step up having made consistent progress
“I’m sure that he will do well in whatever the future holds and we will never hold a player back.”
Meanwhile, Middlesbrough are in the race to sign midfielder Josh Murphy from Championship rivals Norwich City.
The winger, whose twin brother is on the verge of signing for Newcastle United, is the subject of a £4m bid from Garry Monk's side.