The Colombia international was wanted by both clubs this summer but insists he is happy to stay with Fiorentina until the end of the season.
It was reported in August that United was confident of signing the Colombia international this summer in a deal worth up to 37.5 million euros.
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Barcelona now has a transfer ban in place which prevents the club from making new signings until 2016, therefore United could reignite its interest without having to compete with the Spanish club for his signature.
"We had spoken with them [Manchester United and Barcelona] but the truth is that I’ve always remained calm," Cuadrado told reporters.
"I'm happy to stay another season, I'm fine here."