Ashton Gate has already staged several major rugby matches, including last season's West Country derby between Bath and Bristol that attracted a Premiership record crowd of more than 20,000.
It also hosted New Zealand's 1999 World Cup appointment with Tonga, in addition to England A fixtures.
The Wales game, on Wednesday, May 26, has been moved from Cardiff's Millennium Stadium due to pre-scheduled football fixtures.
"Due to the clash with football fixtures and the Millennium Stadium commitment with the Football Association, we have to move our fixture," said Barbarians president Micky Steele-Bodger.
"This year, therefore, we are looking forward to entertaining the Welsh national team and their loyal supporters at Ashton Gate."
The Barbarians will also tackle Scotland on May 22 and England eight days later, with international stars such as Scott Quinnell, Percy Montgomery and Thomas Castaignede among those players who have already accepted invitations.