Hart, whose current deal has two years remaining, helped City win a second Premier League title in three seasons last weekend, responding well to being left out of the first team following a series of high profile errors earlier in the season.
The club is also expected to step-up the search for a new back-up to Hart, with his deputy Costel Pantilimon set to leave the club this summer as he seeks regular first-team football.
Romania international Pantilimon proved his worth to City this season with 18 first team appearances but he is set to leave and Sky sources understand their key target to replace him is Argentinian Willy Caballero, who played under Manuel Pelligrini at Malaga.
The champions are aiming to do their transfer business this summer as quickly as possible and hope to secure their targets prior to the World Cup, and they are poised to offer new contracts to several of the star performers who helped secure two domestic trophies.
England midfielder James Milner will be offered terms to extend his four-year stay at City and Samir Nasri, who scored a vital goal in the final day win over West Ham United but was left out of the French World Cup squad, is also expected to commit his future to the club.
Striker Edin Dzeko is another whose future with the club could be sorted - he hit five goals in his last four appearances of the season to take his overall tally to 28 - and there is also expected to be a new contract for the versatile Aleksandar Kolarov.