Mr. Robinson, the Republican nominee for governor of North Carolina, vowed to stay in the race and sought to deny the report, which also said he had defended slavery in online posts years ago.

Cooper has served two terms as governor of North Carolina, a swing state and key prize for the November election. He is term-limited, and while he was considered to be a contender for Harris’ vice presidential pick, he withdrew his name from consideration before she tapped Minnesota Gov. Tim Walz.