"We will see how he goes tomorrow (Wednesday) and go from there."
However, Green said Thurston had the experience to slot straight into the Queensland side for June 21's Origin II in Sydney even if he missed the Eels clash.
Maroons selectors name their game two squad on Monday.
"He's played enough Origins. He knows what to expect," Green said of three-time Golden Boot winner Thurston.
"And he knows how to get himself ready.
"So as long as his shoulder is 100 per cent fit I have no doubt he will be able to handle the build-up and the preparation for his first game for a while."
Thurston's record streak of 36 straight Origins came to an abrupt halt when he withdrew from last month's game one with the shoulder injury sustained in the May 5 Test win over New Zealand.
Queensland are desperate for Thurston to return after a record 28-4 game one loss to NSW in Brisbane last week.
If Thurston fails to prove his fitness for the Darwin match, Kyle Laybutt will make his debut at five-eighth for the Cowboys with Michael Morgan at halfback.