"We wanted to work with Tumbakovic on a long-term basis but money got in the way as he was given a lucrative offer that he couldn't refuse," Partizan president Dragan Djuric told a news conference.
"We had to let him go and Partizan's doors remain open to Tumbakovic when he returns from China one day."
The 60-year-old Tumbakovic won six league titles in two spells with Partizan as coach and two with Chinese team Shandong Luneng.
He was named Partizan's sports director at the start of this season but reportedly fell out with Djuric earlier this month, after the champions saw their lead at the top of the 16-team first division slashed from 11 points to just four.
On the other hand, Tumbakovic had a sound relationship with Partizan manager Vladimir Vermezovic, who also came under pressure from the fans after two draws and a defeat for Partizan let bitter city foes Red Star back into the title race.
"My departure comes at a bad time but I am a better coach than a sports director and having turned down two offers from China earlier this season, I simply had to take this one because it was nothing short of fantastic," said Tumbakovic.
"I've always been in charge of title-chasing teams, now I will take over a side fighting to avoid relegation and it will be a fresh challenge."
Wuhan are second bottom in the 16-team Chinese first division with only one point from six matches.
Partizan, who are chasing a record sixth successive league title, are likely to name Tumbakovic's assistant Albert Nadj as a caretaker sports director until the end of the season.
Former Chelsea striker Mateja Kezman, a strong fan favourite after winning a league title with Partizan in 1999, has been touted as the most likely long-term successor.
- Sports & Recreation