They surpassed the previous record of 263-5, set by Royal Challengers Bangalore in 2013
Barca's statement added that Dest, 19, had agreed to sign a five-year contract with a release clause of 400 million euros.
Dest, who broke into Ajax's first team from their academy last season and has played three times for the US, will be officially presented on Friday.
The defender will compete with Sergi Roberto for a first team berth after Portugal right back Nelson Semedo left Barca for Wolverhampton Wanderers last week.
($1 = 0.8503 euros)