The Bosnian ace has struggled for first-team action at the Etihad Stadium since his 2011 arrival from Wolfsburg, where he has developed a reputation as one of European football’s top marksmen.
Dzeko is believed to have attracted attention during January, but the Sky Blues held firm to keep him for their season run-in – at which time the quadruple was a possibility.
But now their stance has altered, with the forward due to enter the final year of his contract in under six months’ time.
City are keen to recoup some of their outlay during the off-season, rather than Dzeko leave on a free transfer.
The 27-year-old is understood to be a target for Inter Milan, with the newly Indonesian-backed Serie A giants ready for a summer of big spending.
Interest in his services in Germany is also high, with Borussia Dortmund considering him as a replacement for Robert Lewandowski.
Dzeko started the season as Manuel Pellegrini’s preferred partner for Sergio Aguero, but has since slipped behind Alvaro Negredo in the Chilean’s pecking order.