Dzeko, 32, joined Roma from Manchester City in 2015, and has been something of a goalscoring sensation in the Italian capital.
The Bosnian international has hit a staggering 83 goals in 163 games for Roma, including 39 in the 2016-17 season and 10 so far this term.
Dzeko is out of contract in 2020 though, and Roma may be prepared to sell up in the summer to prevent his value dropping any further.
Calciomercato report that Dzeko has decided he wants a return to the Premier League, and Roma have decided that this summer is the time to cash in for around €15million (£13million).
West Ham are believed to be leading the race for Dzeko's signature, with Roma seemingly ready to allow him to move on at the end of the season.
West Ham boss Pellegrini worked with Dzeko at Manchester City, meaning the duo could reunite this summer, which would be exciting given that he hit 26 goals in his first season under Pellegrini.
With questions over the futures of four West Ham strikers in Marko Arnautovic, Javier Hernandez, Andy Carroll and Lucas Perez, the Hammers could be in the market for a host of strikers, and a proven one like Dzeko may fit the bill perfectly.