Gresham is one of seven restricted free agents across the competition for 2023, joining the likes of North Melbourne defender Ben McKay, GWS swingman Harry Himmelberg and Essendon duo Mason Redman and Darcy Parish in falling under the restricted band.
The 25-year-old, who is in his eighth season at Moorabbin, is among a bevy of Saints players off-contract this year but is said to have hit pause on contract talks with the club for the time being.
While St Kilda are likely to be keen on retaining the South Morang product, they will also have contract calls to make for Hunter Clark, Nick Coffield, Jack Higgins, Zak Jones and Mason Wood, among others.
But should Gresham seek his options elsewhere, the Hawks are seen as a potential landing spot for the free agent, according to The Age's Mark McGowan.
Seen as a strong contender in the potential race for Gresham's signature, Hawthorn will be equipped with a substantial amount of wiggle room within their salary cap following a stronger focus on the National Draft across the past two seasons.
The Hawks are set to assess the available tall forward options within the trade and free agency space, with Gresham seen as a potential conduit for their forward and midfield packs.
The Bunjil Bagora side currently has attacking midfield option Chad Wingard unsigned for next season and will need to make a similar call on his future as they have done with recent high-priced senior figures.
In-form midfielder James Worpel joins Wingard as an uncontracted asset at the Hawks, but has rekindled some of the best form of his career across the opening seven rounds of the season.