The Dragons confirmed on Thursday they were considering granting Graham a release to return to the English Super League.
"The Dragons will consider this option for Graham on the merit a replacement player is secured by the club prior to his release," the club said in a statement.
It's understood Graham only approached Dragons management on Wednesday afternoon and if he did return home it would be to his original club St Helens.
The ball-playing prop played 224 games for the English club before leaving for the NRL and Canterbury at the end of 2011.
In total, he has played more than 400 top-flight games on both sides of the world.
After signing with the Dragons on a three-year deal at the end of 2017, Graham's minutes have dropped in the final season of his contract to just 36 a game.
Given the Dragons want to secure a replacement before his departure, it is likely he will still face the Gold Coast on Saturday afternoon.
There are limited options on the market for St George Illawarra but they will benefit from the fact players outside rival clubs' 17s are not enjoying any game time.
Matt Eisenhuth was linked to Cronulla last month after not playing first grade at the Wests Tigers this year.
Daniel Alvaro is another big-name middle yet to see any NRL action this year, but he remains contracted at Parramatta until the end of next year.