The Bosnia international has been linked with a potential move to the defending Premier League champions over the past week or so.
Dzeko is a more plausible target given his four and a half season spell at Manchester City before moving to AS Roma in 2015.
The 31-year-old scored 50 goals in 130 appearances in the Premier League as the towering striker won the title in 2012 and 2014.
Chelsea have struggled with a lack of creativity in recent weeks following Alvaro Morata’s fitness issues.
Blues head coach Antonio Conte is reluctant to trust Michy Batshuayi despite the Belgian’s tendency to score important goals.
Sky Sports News reporter Sheth suggested that Chelsea and Dzeko still need to agree terms before a deal can be concluded.
“Is the move for Roma striker Edin Dzeko edging closer?” Sheth told Sky Sports News on Thursday morning.
“One source has told us that a deal has been agreed for both Dzeko and left-back Emerson Palmieri. However, there is still work to do on the Dzeko side of it, namely personal terms and length of contract.”
Chelsea signed England international Ross Barkley in a £15m deal from Premier League rivals Everton earlier this month.
The defending champions are in third place in the Premier League table and 15 points adrift of current leaders Manchester City.
The Blues lost 2-1 to Arsenal in the League Cup semi-final on Wednesday night.
Chelsea will take on La Liga giants Barcelona in the Champions League semi-final next month.
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