Green, who represented England on 12 occasions, has claimed that "the next few games could be crucial" in determining which goalkeeper is selected between the sticks in Russia.
"I think there are a couple of tough decisions [for Southgate to make] but I think this is the one where you look at it and you think if Joe Hart had played every game this season, with his experience and being on form, he would be the first choice," Green told Sky Sports News.
"But the way it's gone for him, he's bounced back and he had a great game last week - I'm really pleased for him - but the question mark is still there.
"It's going to be a tough one for Gareth to work out. Maybe he has got ideas already but I think it's something he has to mull over, and the next few games could be critical. I think he said himself in the last international break that the shirt is up for grabs so every performance is key now."
As it stands, Hart, who has long been considered England's number one, will return to parent club Manchester City at the end of the season.