While Real will pursue a deal for Bayern Munich's Robert Lewandowski, Chelsea remain keen on Dzeko. The 32-year-old appeared set to move to Stamford Bridge in January, along with then-team-mate Emerson Palmieri, according to Harris.
Palmieri completed a move to west London, but Roma chief executive Umberto Gandini vetoed the winter deal for Dzeko, per the article.
Since then, the veteran striker has been in prolific form, recently scoring Roma's winner against Shakhtar Donetsk in the UEFA Champions League last 16. Overall, Dzeko has helped himself to 17 goals and five assists.
The former Manchester City man is a classic No. 9, a natural target man who can boss defenders in the air and play with his back to goal. He's also a cooler finisher than many forwards his size.
Chelsea haven't been able to find the right formula up front, with manager Antonio Conte unconvinced by his options through the middle. Alvaro Morata moved to Stamford Bridge in a deal worth a club-record £60 million last summer.
Despite a fast start to life in the Premier League, injuries and inconsistency have blighted the Spain international's debut season. Those issues led to Chelsea signing Olivier Giroud from London rivals Arsenal, with Michy Batshuayi joining Borussia Dortmund on loan as part of the deal.
Like Dzeko, Giroud is a throwback centre-forward who can act as the focal point up top. The Frenchman has so far been consigned to limited action, though, making just two starts in the league since joining Chelsea.
Adding Dzeko alongside two strikers as similar as Morata and Giroud would seem like a questionable move.
It's especially questionable since Conte still sounds confident in Morata's ability, per ESPN.com's Graham Hunter: "To hear Conte say in Barcelona the other week that Morata is a great player in the making and that he's been impressed by the 25-year-old's efforts to win his place back was hugely encouraging."
Dzeko also told Bosnian source Klix (h/t TalkSport) in February he's content in the Italian capital: "In the end I'm happy to stay at Roma, because Rome has become my home and an indelible part of my life."
It seems likely Dzeko has missed his chance to join Chelsea. The Blues would be better served looking for a more mobile striker to provide Morata with support.