Swansea's Lukasz Fabianski has dismissed rumours of a move to Roma insisting he is happy in the Premier League.
Head coach Kenny Jackett has not given up hope of keeping Bakary Sako at Wolves beyond the end of the season.
Dzeko has impressed during City's pre-season campaign, scoring in both Barclays Asia Trophy games against South China and Sunderland and then putting two past AC Milan in the Audi Cup in Munich.
There had been speculation over the Bosnian's future but new boss Pellegrini has spoken of his faith in the player, who has not consistently impressed since his £27million arrival from Wolfsburg in 2011.
Dzeko, 27, made clear his frustration at often being restricted to substitutes' roles last season but, despite sometimes being criticised for his finishing, ended the campaign with 15 goals.
Sergio Aguero and new signings Stevan Jovetic and Alvaro Negredo will be challenging for places up front next term but Dzeko seems ready for the challenge.
He told City TV: "I feel different, better. I feel the I have more confidence, the manager gives me more confidence. That is an important thing."
Pellegrini has referred to Dzeko as a "top striker" and backed him to reproduce the form that made his name in Germany.
When asked if that support was important, Dzeko said: "Of course it is. I think you can see on the pitch how I play.
"I feel different, also because you know someone who is important has trusted you. It is good, for now it is looking good."