Barcelona goalkeeper Claudio Bravo is set for a medical at Manchester City.
England international Hart, 29, began the campaign as second choice to Willy Caballero at City. Guardiola, who took charge in the summer, has major reservations about Hart's ability to play as the kind of 'sweeper-keeper' his preferred system demands.
It was expected Hart would leave the club before the transfer window closes on 31 August, but an immediate transfer is no longer certain. He does not want to move to Sevilla or Borussia Dortmund, both of whom are reported to be interested in signing him.
Everton, which seemed a potential destination late last week, appear to have cooled their interest - partly due to Hart's wages, which are beyond their current structure, and the early-season form of new signing Maarten Stekelenburg.
England manager Sam Allardyce has said Hart's situation is "a concern" but that he would pick the player in his squad for the World Cup qualifier against Slovakia on 4 September.
Hart trained with Cabellero at the City Football Academy on Tuesday morning.
Yaya Toure, Wilfried Bony, Samir Nasri and Eliaquim Mangala, who have also played no part for City this season, were among the 33 players who trained across two pitches.