The report contains a bogus poll indicating that President Uhuru Kenyatta will win August's election.
The BBC has urged Kenyans to verify stories by visiting the BBC website.
A recent survey suggested that 90% of Kenyans had seen or heard false news in the run-up to the poll.
Many of the people surveyed felt that some news items had been deliberately misleading.
It also found that while traditional media remained the most trusted source of information, large numbers of people got their news from Facebook and WhatsApp.
The origin of the fake BBC news story is not clear but it began to be shared on WhatsApp on Friday morning.
A fake report made to look as if it came from CNN has also been circulating.
The presidential election on 8 August will see incumbent President Kenyatta run against seven candidates including his main challenger, opposition leader Raila Odinga.
A fake video report about Kenya's election that is made to appear as if it is from the BBC's Focus on Africa programme has been circulating on social media.