The pair worked together at Athletic Bilbao and have exchanged praise through the media this season.
With Herrera coming off contract in June, there has been hope from Valverde of bringing him to Barca next year. There were plans to offer the midfielder a pre-contract deal.
However, Marca says Barca have learned from intermediaries that United will trigger a 12-month extension clause in his current deal.
It's suggested Barca haven't ruled out trying again, but it will now cost them.