According to local media, Shenhua are currently locked in talks with the Chelsea forward and are optimistic a deal is not too far away.
It would represent a huge double coup for Shenhua, who persuaded Drogba's former Chelsea team-mate Nicolas Anelka to make the move earlier this year.
"Everything is going well so far though there have been a few obstacles. They never turned us down, which is good for us," club board member Zhou Jun told The Shanghai Daily.
With Anelka having recently signed with the Chinese side, Zhou believes the odds of signing the 33 year-old has shortened dramatically.
"It's like making a blockbuster film. When superstars like Jackie Chan, Jet Li and Zhou Xun have confirmed their participation, other actors and actress will be confident of the movie."
Details of the negotiations have yet to be disclosed but Drogba is expected to join Anelka as one of the highest-paid players in China if he decides to make the move.