Herrera has only played a bit-part role under Jose Mourinho this season, making just 24 Premier League appearances with half of those coming as a substitute.
The 28-year-old is understood to be keen on a return to Spain in search of more regular first-team football, with former club Athletic Bilbao initially thought to have been the favourites for his signature.
However, El Gol Digital claims that Barcelona have now entered the race with a £22m bid which United are willing to accept.
Herrera's first-team chances are likely to be limited at Camp Nou too, but the move would see him reunited with former Athletic boss Ernesto Valverde.
The player joined United in 2014 and has since gone on to make 158 appearances for the club across all competitions, scoring 17 goals.