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.

Thirty-six year old Marie Low died in Ninewells Hospital after an incident in the city's Ballantrae Terrace.
She was found with serious injuries at about 14:55 on Sunday.
A 30-year-old is expected to appear at Dundee Sheriff Court on Tuesday. Police Scotland said there would be a continued police presence in the area. Tributes have been paid online to Miss Low by friends and family.
Her daughter, Shanice Low, wrote on Facebook: "R.I.P mummy I love you millions I can't believe your gone. I'm going to make you proud love you loads mummy."
Her brother, Kevin Low, added: "Love you sis - gone but never forgotten."
Several other friends have changed their profile photos on social media to images of Marie.
Her sister, Denise Low, said: "It was our dad's birthday yesterday - we can't believe this has happened.
"My sister was strong and beautiful and will be sorely missed by us all.
"She was one of the best people I knew.
"She was a very, very strong woman. She always looked great."
One friend has also started a crowdfunding page.
Newly promoted Premier League side Hull City are interested in signing Reading and Wales defender Chris Gunter.
[NEXT_CONCEPT]
A woman has been charged following the death of another woman in Dundee.