Joshua also holds the IBF and WBO titles and his camp have been in negotiations to set up a unification fight with WBC champion Deontay Wilder.
But a fight with the American is yet to be finalised and the WBA wants a bout against Russian Povetkin to be set up.
Povetkin knocked out David Price in March to become the mandatory challenger.
If Joshua fails to fight Povetkin he could be stripped of his wba belt.
"Today the WBA requested an answer from Anthony Joshua's team regarding his fight with the mandatory challenger Alexander Povetkin," said WBA president Gilberto Mendoza.
Joshua's promoter Eddie Hearn said his fighter was "frustrated" over the time it was taking to reach an agreement with Wilder's team.
"Anthony is frustrated. We sent the contract nine days ago. Nothing makes sense here at all. They know we're under pressure from the WBA. It feels like a big game.
"He wants to fight Wilder but at the same time we are not going to lose a belt to the WBA.
"Right now it looks like Povetkin is next but anything can happen in this sport."
Courtesy of the BBC