Hart has made 32 appearances for his hometown club since he was handed his league debut last season and is 6ft 5in tall.
Goalkeeping coach Tim Flowers has been to watch the highly-rated 18-year-old on several occasions, with Stuart Pearce watching him in the Carling Cup tie with Sheff Utd in September, when Hart's team were beaten on penalties.
Young keeper Kasper Schmeichel has recently signed a new contract with City but Nicky Weaver and Geert de Vlieger are both due to become free agents in the summer.
Meanwhile, talks are still progressing with Spanish winger Albert Riera over a loan move, despite Espanyol announcing last night a deal had been done.