The 34-year-old helped the Blues win the Champions League in Munich to bring the curtain down on a trophy-laden spell in west London.
Ivory Coast international Drogba - signed from Marseille for £24million in 2004 - is now expected to join up with former Blues team-mate Nicolas Anelka at the Chinese Super League club on what would be a lucrative free transfer.
New Shanghai Shenhua manager Sergio Batista told reporters: "The people from the club are doing the best things to get Drogba here.
"I think every coach in the world wants to get Drogba on the team and I hope everything is okay."
Former Argentina manager Batista maintained there was no issue with Anelka, who had taken temporary charge as player-coach after former manager Jean Tigana was sacked in April.
"It's another moment. He's a player. I'm open to hear Anelka and all the players on the team speak about what's happening on the field," Batista, 49, said.
"Obviously, the final decision is from the coach.
"Anelka can give to all the players his experience, on the field and off the field.
"I don't want Anelka to play only to score goals. He is one of the best players in the world, but I want to keep him back on the field to play with his partners on the team."