Deco sees the PSG coach as the ideal replacement for Xavi, who will step down at the end of the season.
Cadena SER says Deco has already chosen and contacted the former Bara coach, who has a deal with PSG only to 2025.
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Enrique took charge at PSG last year.
The alternatives for the bench at Barca are Roberto De Zerbi from Brighton and the German Hansi Flick, who is currently a free agent.
But Deco wants to bring his former Barca teammate Enrique back to the Nou Camp for a second spell in charge.