Brown is keen to strengthen his back-line ahead of the new Premier League season, and he has identified Gardner as a transfer target for the summer.
The 27-year-old centre-back is expected to soon leave White Hart Lane after failing to secure a first-team position under manager Juande Ramos.
Gardner is now back to full fitness, and Brown has revealed his interest in purchasing the former Port Vale man - who is also a reported target of fellow promoted side Stoke City.
"There's interest from us," Brown told the Hull Daily Mail.
"The lad's got a proven Premier League pedigree and is available at the right transfer fee.
"How far we take the interest will be decided more than anything on if he wants to come up here. But we have made it clear to him and his club that we are interested."