The Nagata Conjecture is one of the most intriguing open problems in the area of curves in the plane. It is easily stated. Namely, it predicts that the smallest degree d of a plane curve passing through r ≥ 10 general points in the projective plane P 2 with multiplicities at least l at every point, satisfies the inequality d > √ r · l. This conjecture has been proven by M. Nagata in 1959, if r is a perfect square greater than 9. Up to now, it remains open for every non-square r ≥ 10, after more than a half century of attention by many researchers. In this paper, we formulate new transcendental versions of this conjecture coming from pluripotential theory and which are equivalent to a version in C n of the Nagata Conjecture.