Dzeko, 33, made his name with Manchester City following a 2011 move from Wolfsburg, and ended up winning two titles with the club – one under Pellegrini.
The Bosnian hit 72 goals in 189 games for City before leaving for Roma in 2015, where he has continued to score goals, finding the net 86 times in 171 outings.
However, Dzeko's contract expires in 2020, and with Roma reluctant to offer him a new deal given his age, it seems that he could be on the move this summer.
Sky Sports now claim that Dzeko has been offered to West Ham, and it may well be tempting for Pellegrini to strike a deal given their prior relationship at City.
In fact, The Express recently reported that West Ham were already preparing to hand Dzeko a huge £165,000-a-week deal in order to make the move to the London Stadium.