The 27-year-old signed a new contract with the Championship club in February, keeping him at the Madejski Stadium until 2018.
But the new deal includes a release clause of £1.2m for Gunter, who helped Wales reach the Euro 2016 semi-finals.
Hull's interest is believed to be on hold as they search for a new manager following the departure of Steve Bruce.
Injuries have left the club with only 13 fit players, and they said they would look to strengthen the squad in the "coming weeks".
Right-back Moses Odubajo is one of several important players currently injured, and Gunter is regarded as a potential replacement.
The former Cardiff City and Tottenham defender played in all six of Wales' games at Euro 2016 as they reached their first semi-final at a major tournament.
The progress of Chris Coleman's side in France meant Gunter was forced to miss his brother Marc's wedding.
Newly promoted Premier League side Hull City are interested in signing Reading and Wales defender Chris Gunter.