Tarrant is seeking a return to Melbourne, preferably with his original club Collingwood, but the Hawks have been strongly linked to him.
Hawthorn general manager player personnel and strategy Chris Pelchen says the club would love to have Tarrant on board.
"There's genuine interest there, there's no doubt, but Paul Connors his manager has explained he'd like to go to Collingwood and we've respected that at this point in time," Pelchen said as he arrived at Etihad Stadium for the commencement of trade week on Tuesday morning.
Pelchen said it was unlikely the Hawks would be part of any early trades.
"It wouldn't be this early. We've had discussions with all clubs except St Kilda to this point and it would be fair to say it's very preliminary," he said.
"We might know a bit more in about six hours, we've got a couple of clubs we want to ask about but it better remain private until later in the day."
Pelchen said there was no one specific player type the Hawks were after.
"There are probably several. Obviously to improve our list is the main objective. We're looking at midfielders and obviously backmen, that is widely known," he said.
"We haven't discounted any player position provided that improves our club."
