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Thinking about the coherent sheaf case, this resembles a question of Totaro. Totaro asks whether, for any finite type variety X, and any coherent sheaf E on X, there is a vector bundle V on X with a surjection V --> XE. This question is open and appears to be quite difficult.

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Thinking about the coherent sheaf case, this resembles a question of Totaro. Totaro asks whether, for any finite type variety X, and any coherent sheaf E on X, there is a vector bundle V on X with a surjection V --> X. This question is open and appears to be quite difficult.