Citing sources, the Independent said United had not received an official approach of any kind for the Spain international, who was voted the Red Devils' player of the season.
Barcelona are expected to announce Ernesto Valverde, who stepped down as Athletic Bilbao coach this week, as their new coach following the club's Copa del Rey final showdown with Deportivo Alaves on 27 May.
Valverde worked with Herrera for one season at Athletic Bilbao before the midfielder joined United in 2014 and is understood to be an admirer.
However, the Red Devils are reported to be keen to offer a new contract to Herrera after the key role played by the midfielder in the club's Europa League winning campaign.
The 27-year-old was named man of the match in United's 2-0 win over Ajax in the Europa League final in Stockholm. He dedicated the award to the victims of a terror attack that killed 22 people and injured dozens more following a concert at Manchester Arena on 22 May.
"We were really, really sad," Herrera was quoted as saying by Sky Sports. "When you think about life and football, you have no choice.
"It was difficult to train yesterday, really difficult for us but the manager told us that the only thing we could do was win this for them and that's what we've done.
"It's just a football game, it's just a trophy, but if we can help or if we can support them with this just one per cent it's enough for us."
Herrera made 31 league appearances for United in the 2016/17 season, scoring one goal. He has a year left on his contract with the club, although the Red Devils have the option to extend it by an additional year.