Diedhiou was subjected to the abuse online after missing a penalty in his side's 1-0 defeat by Swansea City on Saturday.
Connor Roberts gave Swansea the lead on the stroke of half time before Diedhiou, 27, failed to level proceedings as his 69th-minute spot kick cannoned off the post.
After the game, the Senegal international posted a screenshot of a message with three banana emojis, which he received from someone on Twitter. Diedhiou commented, "Why??"
"We stand with Famara Diedhiou and anybody who has to suffer racism in any shape or form. There is no excuse or justification for this disgusting behaviour," Bristol City said in a post on Twitter.
The incident comes shortly after Crystal Palace's Wilfried Zaha and Sheffield United's David McGoldrick were subjected to similar abuse on social media.
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The 24-year-old Rangers striker was streaming on Instagram live, during which other users are able to post comments, when the incident occurred.
A Police Scotland spokesperson said: "We have been made aware of offensive comments online and officers are investigating. We take all allegations of online abuse seriously and will thoroughly investigate such matters."
Additional reporting from Reuters.