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 chain of people passed a light to each other, finishing at the cathedral.
As part of the annual Peace Festival in the city, the cathedral was lit up in blue, white and red to mark Friday's attacks in Paris.
The event was led by the Bishop of Coventry and the Duke of Kent.
The human chain idea came from a similar event held annually in Dresden to commemorate the allied bombing of February 1945, which destroyed the city's centuries-old baroque city centre.
BBC Coventry and Warwickshire was broadcasting the event live until 21:00 on Saturday.
As part of the ceremony, the Saint Michael's Singers also performed Mozart's Requiem and Solemn Vespers with the Parliament Choir and the English Symphony Orchestra.
Ann Lucas, from Coventry City Council, said: "With each passing year the relevance of the human tragedy of the event and how Coventry rallied to become the city of peace and reconciliation grows in significance.
"Whilst remembering the horror of 1940 is vitally important, it's also key we view this as an opportunity for looking forward and ensuring that people and communities come together as a city to work towards a place we can all be proud of."
In Dresden, an estimated 22,700 to 25,000 people died as a result of a series of raids by British and US bombers during World War Two.

The 37-year-old suffered the injury on 4 June during the first innings of Sussex's County Championship Division Two win over Worcestershire.
It was only the Australian's second Championship appearance of the season after he missed three of Sussex's previous four games with a knee injury.
Magoffin took 5-51 in his only other game as Sussex lost at Nottinghamshire.
Sussex are currently fifth in the Division Two table, with two wins from their opening five matches.

Adam King struck either side of the break in the 2-0 win over Reading at Liberty Stadium.
In March, the Swans' development squad drew with Wolverhampton Wanderers to take the league crown.
The cup final win meant the young Swans celebrated 31 wins across all competitions in 2016-17.
The youngsters' success came after Swansea's senior team survived in the Premier League after Crystal Palace beat Hull 4-0 on Sunday for the latter to be relegated.
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.
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Hundreds of people have taken part in a ceremony to mark the 75th anniversary of the Luftwaffe bombing raid on Coventry, which left 568 people dead and most of the city centre in ruins.
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Sussex seamer Steve Magoffin will be out for six to eight weeks after suffering an Achilles tendon injury.
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Swansea City Under-23s have won their season's double after adding the Premier League Cup to their Premier League Two title.