The Senegal international was reported to have signed a three-year deal worth £11.3m, Xinhua news agency said on its website, after the Shanghai club confirmed the 30-year-old would be joining.
The figure would eclipse the previous CSL record of £10.6m paid in January by champions Guangzhou Evergrande for Brazilian Ricardo Goulart.
The Shanghai club did not confirm the figure, Xinhua added, referring to a “possible record transfer fee”.
Ba, who will undergo a medical on Sunday, made the switch to China from Turkish side Besiktas, where he scored 27 goals in 44 games last season.
The 30-year-old striker enjoyed spells in England with West Ham, Newcastle and Chelsea before joining Besiktas and had been linked with a return to the Premier League.
But he will instead line up alongside former Everton star and Socceroo Tim Cahill and Mali international Mohamed Sissoko, who arrived in China from La Liga side Levante after spells at Paris Saint-German and Liverpool.
Ivory Coast veteran Drogba – who left Chelsea on a free transfer at the end of last season – and his former teammate at Stanford Bridge, Nicolas Anelka, both previously played for Shanghai.