The deal is estimated to be worth £25m a year.
The Qatar Foundation is a non-profit organisation, which primarily runs education projects aimed at helping the country and other parts of the Middle East.
Barcelona has previously refused commercial shirt sponsorship.
The move by the club could free up transfer fees for manager Pep Guardiola, helping him lure new football stars to join the likes of Lionel Messi and Andreas Iniesta.
The Qatar Foundation will share the shirt space with Unicef, after the children’s charity became the club’s first sponsor in 2006.
It is thought that Barcelona will look to give the two logos equal billing, but should that be prove difficult, the Unicef logo will be the dominant one.
The Qatar Foundation, which was set up in 1995, is chaired by Sheikha Mozah Bint Nasser Al-Missned, the wife of the Emir who was central to Qatar’s successful bid to host the 2022 World Cup competition.