The 22-year-old centre-half, who spent the second half of last season on loan at Wigan in the Championship, only signed for Hull from Sheffield United last summer for £2.5million.
Maguire barely got a look in as Hull were relegated from the Premier League last season, making just three appearances in the league as a sub and two starts in cup competitions.
But he made 16 first-team appearances for Wigan as they were relegated to League One.
Bristol City have denied making a bid for the player, who wants to play regular first-team football, but sources close to the Yorkshireman understand they are very interested and have lodged a bid to Hull.
Maguire began his career with Sheffield United and made over 150 appearances for his home-town club, scoring 12 goals.