The England number one has been linked with Arsenal, Liverpool and Chelsea in recent weeks, but the word coming out of the Etihad Stadium is the 29-year-old will have to forget all about joining a club the Citizens consider a rival for top four.
That would appear to limit Hart's options, who still has two years left to run on his £130,000-a-week City deal.
Hart has impressed during his season-long loan spell at Torino and it is understood his camp are aware of the stance his parent club holds.
A permanent stay in Turin has not be ruled out, with City open to a cut-price £15m deal if Hart remains on the continent.
Pep Guardiola could be in the market for two new goalkeepers this summer although there is little hope of an emotional recall for the former Etihad favourite.