Murtaza Ahmadi, now aged seven, went viral in 2016 after being photographed wearing a homemade Messi shirt, fashioned out of a plastic bag.
He later met his hero in Qatar.
His family say they have now abandoned their home in Afghanistan, after receiving threats from the Taliban.
They were living in the south-eastern Ghazni province - which militants have been targeting - and have escaped to the capital, Kabul.
They previously sought short-term refuge in Pakistan in 2016, but returned when they ran out of money, according to AFP news agency.