Having signed the St Helens Englishman on a three-year deal from 2021, Canterbury are hoping he can secure an early release from the Super League club.
The Bulldogs still have up to four spots available on their list and want Thompson to occupy one of them.
“We’re looking at what’s happening in Super League and what opportunities there may be to bring Luke here as early as possible,” Bulldogs chief executive Andrew Hill told The Sydney Morning Herald.
“There’s lots of hurdles for us to overcome at this stage and there has been no real progress on that opportunity.”
The NRL is still yet to decide how far back it will push the June 30 transfer window, which will affect Thompson’s transfer.
The Bulldogs are also eyeing wantaway Raiders star John Bateman.