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The 21-year-old injured his left hamstring during the early stages of the second half in Barca's 2-1 UEFA Europa League Round of 16 second-leg win over Galatasaray on Friday (AEDT).
Dest has undergone a scan and Barca confirmed on Saturday (AEDT) that he will not return to action until after the international break.
As a result, Dest has been replaced by Arminia Bielefeld left-back George Bello in United States' squad for their qualifiers against Mexico, Panama and Costa Rica.
Dani Alves is expected to slot in for Dest in a straight swap for the LaLiga trip to Madrid, but the veteran full-back is not registered for the UEFA Europa League.
Barca will face Eintracht Frankfurt in the quarter-finals of that competition, with the first leg scheduled for 8 April (AEDT).
Ajax academy product Dest has featured 20 times for Barca this term – the 10th most of any player – and has been capped 17 times by United States.