The England international has not played for Manchester United since December, following a disagreement with head coach Ruben Amorim, and is among several senior players not allowed to train with the first team.
Barcelona's interest in Rashford was publicly confirmed by manager Hansi Flick and sporting director Deco, after their attempts to sign Luis Diaz and Nico Williams were unsuccessful.
Meanwhile, Manchester United are progressing with other transfers, including Bryan Mbeumo's medical after agreeing a fee with Brentford, and are looking to sell several other players.