JOHANNESBURG—With Africa’s most poverty-stricken economies pushing for debt relief as they struggle with the fallout from the global coronavirus pandemic, a tussle is brewing between developed nations, private investors and the continent’s largest creditor: China.

Wealthy countries from the U.S. to Japan—watching their own economies lurch toward recession—are loath to forgive African debt if they think the money will indirectly support Chinese creditors, including the government, banks and contractors. At the same time, Beijing...