The Islamist group says it has taken "complete control" of the AU camp and killed more than 60 Kenyan soldiers.
Residents told the BBC that al-Shabab had raised its flag at a camp for Kenyan troops and said insurgents had paraded bodies through the town.
But Kenya's army spokesman said it was a nearby Somali base that was stormed and Kenyan troops counter-attacked.
The number of casualties on both sides was not known, Kenyan military spokesman Col David Obonyo said in a statement.
He told the BBC that reports of high Kenyan casualties were "normal propaganda from al-Shabab".
Kenya contributes more than 4,000 troops to the 22,000-strong AU force that is in Somalia helping the UN-backed government battle al-Shabab, which is part of al-Qaeda.
An el-Ade resident told the BBC by phone that he had heard a loud explosion at about 05:30 local time (02:30 GMT), followed by heavy gunfire.
"We then saw an al-Shabab fighter in the town.
We also saw Kenyan soldiers who were fleeing from the camp.
"At the moment the camp is in the hands of al-Shabab.
We can see military cars burning and dead soldiers all over the place.
There are no civilian casualties but most people have fled the town."
Al-Shabab is keen to stress - both to the foreign forces in Somalia and would-be defectors within its ranks enamoured by so-called Islamic State - that it is a force to be reckoned with.
It is true that the group exaggerates the scale of its attacks but whether or not the death toll from el-Ade attack is as high as claimed, the African Union mission (Amisom) will have to recognise it still has a long way to go to gain full control of the country.
It says a lot for the insurgents to be daring enough to strike at the heart of an African Union base and engage troops in hours of combat.
They have done this before - in two major attacks targeting the Somali military last year, dozens of Burundian troops and Ugandans soldiers were killed.