The pair previously worked together at Manchester City and the Chilean is understood to want a reunion in London.
The Express says that West Ham failed with a bid to land the 33-year-old in January but will knock on the Serie A side’s door again this summer.
Pellegrini’s side are said to be willing to offer a two-year deal worth £165,000 per week to convince the Bosnian to return to England.
Everton are also interested in a move for the two-time Premier League winner, but Dzeko’s preference to live in London could give the Hammers the edge.
The former Wolfsburg striker has scored seven goals in 26 Serie A games for the Giallorossi this season.
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