Thompson, who is yet to play this season due to injury, is eager to remain in the NRL but runs off-contract at the end of the season and is unlikely to be re-signed by the Bulldogs.
The English international is in the final year of his contract, which purses him $800,000 a season, but he has struggled to live up to expectations at Belmore.
Signing Thompson would increase the Dragons' forward stocks, and he would slot right into the starting team. The news, revealed by News Corp, comes days after Zac Lomax recommitted himself to the club after there were reports stating he had permission to look elsewhere.
It is understood by the publication that Flanagan and Thompson had a face-to-face meeting recently regarding his future.
After undergoing surgery in the pre-season for a foot injury sustained at training, Thompson is on the mend and is aiming to play this season before it ends. Aged 28, the Dragons aren't the only club interested in his services, with multiple overseas clubs hoping they can entice him with a return back to England.
St George has also recently been linked to Gold Coast Titans duo Tino Fa'asuamaleaui, David Fifita, and Papua New Guinea international hooker Edwin Ipape- presently playing for Leigh in the Super League.