Fati has reportedly emerged as an alternative target to Borussia Dortmund star Jadon Sancho for United as the Premier League giants eye attacking reinforcements.
There has been talk of United making a £100million (€111m) bid for 17-year-old Barca sensation Fati, who only made his senior debut in August.
However, Vilajoana dismissed the speculation over Fati's future at Camp Nou.
"For us, there's no story with Ansu, we have never spoken about selling him," Vilajoana told Sport.
"We won't use players from the academy to make cash. In principle, we have no intention of selling the players we believe will make the first team."
Vilajoana added: "I think he's happy. He will be an important player in the coming years."
Prior to the coronavirus pandemic, Fati had scored four goals and supplied an assist in 16 La Liga appearances for Barca.
Fati has played 24 matches in all competitions this season, scoring five goals.
Back in February, Blaugrana coach Quique Setien praised the talented youngster after he scored a brace in the 2-1 win against Levante.
"I think that Ansu is a boy that is emerging, that we all know that has great potential," Setien told a news conference.
"You have to understand that you have to give him peace of mind and confidence to exploit what he has inside.
"Apart from the two goals, what I give credit is the work he has done in defence. He has worked hard. He has given us solutions."
Fati was born in Guinea-Bissau but has decided to represent Spain at an international level and already has caps for the nation's under-21 team.
He honed his talents at his local Spanish club Herrera before moving to Sevilla's academy at the age of eight.
But after only two years at the Seville club, Fati was picked up Barcelona's La Masia academy and signed a professional contract prior to the current season.
La Liga leaders Barca will return to action with a trip to Real Mallorca on Saturday.