Barcelona goalkeeper Claudio Bravo is near to completing a move to City to seemingly replace the England international.
Hart is said to be keen to leave City in search of regular first-team football - the 29-year-old does not appear to be in Pep Guardiola's plans and the Spaniard has not used him in any of the club's official games so far.
Hart is reportedly interested in a move to Sevilla in Spain's La Liga but such a deal would mean City heavily subsidising his £140,000-a-week salary.
However, Everton could also emerge as a strong candidates to land his signature. After not compromising on their asking price for John Stones, the club would be expected to pay a loan fee for Hart plus his wages in full.
Toffees coach Ronald Koeman has also hinted he would consider a move for the shot-stopper - Everton have three keepers although one of their options is 19-year-old Mateusz Hewelt.
Meanwhile, Bravo's deal to join City is apparently signed and sealed with the 33-year-old having already told his Barcelona teammates that he is joining Guardiola's side.
Hart has made 347 appearances for the Citizens and has won the Premier League and League Cup twice, as well as the FA Cup once.