The Hexham-born former Newcastle United goalkeeper said last week he would love to return to Scotland wearing an England shirt.
The Manchester City goalkeeper will not travel with the squad to face Gordon Strachan's side on Tuesday, and either Fraser Forster or Ben Foster will be given the opportunity to stake a claim for a regular place.
Hart's position for both club and country had come under threat earlier in the year after a number of high-profile mistakes.
Roy Hodgson (talking to ITV): "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."
Forster has just two caps to his name to date, the second of which came as a late substitute in a pre-World Cup friendly draw against Honduras.
He has been in superb form this season, helping take Southampton to second in the Premier League behind Chelsea.
GIVE FRASER HIS DREAM
Former Toon goalkeeper Fraser Forster would love to represent England against auld enemy Scotland.
The match will be the first time England have travelled north of the border since 1999.
The 26-year-old keeper collected three Scottish league titles during a four-year stay with Celtic.
Fraser Forster (talking to Sportinglife): "There's no such thing as a friendly between Scotland and England.
"It had been a great start to the season for me personally, and a fantastic start for us as a club.
"It's just a case of turning up and seeing what happens, really. It is up to the manager what team he goes with.
"It will be a very special occasion, definitely not quiet.
"It's a stadium with a fantastic atmosphere and for a Scotland v England match it will be a very different experience to anything I've ever experienced.
"It will be something different to what I've experienced but it'll be a fantastic game.
"It is all about Slovenia, really, and looking at them first and foremost."