The forward struggled for first-grade opportunities in John Morris’ side in 2020, playing just three NRL games.
Off-contract at the end of 2021, the Wests Tigers are also reportedly interested in the 23-year old.
Cronulla granted Magoulias permission to talk to rival clubs in July despite still having another year to run on his contract.
The Dragons wants Magoulias in the Red V in 2021 despite an understanding that he could be prepared to stay at the Sharks and work hard to crack into their first-choice side.
The Dragons are hopeful that the lure of a two-year deal and more regular NRL opportunities will be enough to convince him to make the move, with new coach Anthony Griffin seeing value in his flexibility to play multiple positions.
Magoulias has played five NRL matches since making his debut in 2018.