Fans of Mr. Coates accused Mr. Dokoupil of bias, with one writer for Vox calling his questions “hostile, combative and rude.” Others took a more sanguine view, including a Washington Post reporter who wrote that the conversation had been “impassioned but calm” and had brought rigor to the typically breezy realm of morning TV.

Late last week, a group of CBS News employees approached executives with their concerns about Mr. Dokoupil’s handling of the interview, according to two people with knowledge of the events, who requested anonymity to share internal discussions.

Mr. Dokoupil met for an hour with members of the CBS News standards and practices team and the in-house Race and Culture Unit, which advises on “context, tone and intention” of news programming. The conversation focused on Mr. Dokoupil’s tone of voice, phrasing and body language during his interview with Mr. Coates, one of the people said.