The Teessiders made bids of £4million and £5million for the former Sheffield United youngster during the summer but were knocked back on each occasion by the Tigers.
Hull were keen not to let Maguire leave at the time as they were short of centre backs due to an injury crisis, but might be more open to a sale in January.
Boro boss Aitor Karanka has reportedly been encouraged to try again for Maguire due to Hull's failure to get the 23-year-old to sign a new contract, with his current agreement set to expire at the end of the season.
Maguire is eager to be playing regularly at Premier League level at this stage of his career but has made just seven top-flight appearances for Hull this season.
The defender has previously said he wants his future sorted before the January transfer window opens and therefore could be open to a move to the Riverside if a new deal at the KCOM Stadium has not been agreed.
Middlesbrough last week