Hodgson's men remained unbeaten in Euro 2016 qualifying on Saturday, coming from behind to beat Slovenia 3-1 at Wembley.
Hart allowed the first goal of the game, Liverpool midfielder Jordan Henderson leaving his keeper with little chance as he headed into his own net in the 58th minute before England hit back with three goals in 13 minutes - a penalty from centurion Wayne Rooney and a brace by Danny Welbeck.
Hodgson said he wanted to look at his back-up goalkeepers for their friendly with Scotland in Glasgow on Tuesday.
"Leighton Baines was one but he got released anyway," Hodgson said, of the Everton man who departed the squad on Friday due to injury.
"The other of course is Joe Hart because I have still got the other two goalkeepers and they will come with us up to Scotland and one of those will play - Joe will go home.
"They are the only ones, obviously we have got players that I would like to see so there is a potential for me to change things but it is not going to be a question of sending the bulk of these players home because Scotland is a tough game and we want to go up there with a big team."
Foster played for England at the World Cup, keeping a clean sheet in their 0-0 draw with Costa Rica - when qualification was already out of reach for Hodgson's men in Brazil.
Forster, who is enjoying a fine season with in-form Southampton, has twice been capped at international level and has good pedigree playing at Tuesday's Celtic Park venue - having made 140 league appearances for the Scottish champions from 2010-14.
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