Maguire was the subject of a £4 million bid from Boro last week, but Hull turned down the offer due to a number of injury issues which have left them with a depleted squad ahead of the new Premier League season.
Boro are also short of centre halves, following injuries to Daniel Ayala and Bernardo Espinosa, which appears to have prompted a pursuit of Maguire, who joined Hull in 2014.
However, the Teessiders have suffered another setback with Hull rejecting an improved bid of £5 million for the 6ft 4ins defender.
Hull are struggling for numbers
Maguire, who joined Hull from Sheffield United for £2.5 million, is believed to be keen on the move after playing just 22 league games for the Tigers in the Championship last campaign.
Hull are still looking for a new manager following the departure of Steve Bruce 12 days ago, with the former West Ham and Watford manager Gianfranco Zola the favourite to take over at the KC Stadium.
Whoever takes over will hope to bring in more players before the transfer window shuts in less than a months’ time. Hull have made just two signings since they were promoted back to the Premier League and were said to only have '13 fit senior players' when captain Michael Dawson was ruled out for three months with a knee injury a couple of weeks ago.
That could open the door for Maguire who was often seen as a back-up last season – behind the experienced pairing of Dawson and Curtis Davis.
Boro still want a centre half
Despite making nine new signings this summer, Boro also appear a little light at the back and it is unclear whether Ayala or Espinosa will be ready for their Premier League opener with Stoke City next week.
Ben Gibson and 18-year-old Dael Fry played the full 90 minutes in Boro’s 3-1 friendly victory over Aston Villa on Saturday.
Following the game Karanka told the Gazette that he is still searching for another centre half before the season kicks off on August 13.