Distributing elements across a list of lists

This works:

Transpose[{v, #}] & /@ u

{{{0, 1}, {4, 3}}, {{0, 2}, {4, 6}}, {{0, 3}, {4, 9}}}

Transpose /@ Tuples[{{v}, u}]
Transpose @@@ Table[{j, i}, {i, u}, {j, {v}}]
Transpose /@ Partition[Riffle[u, {v}, {1, -2, 2}], 2]

(*{{{0, 1}, {4, 3}}, {{0, 2}, {4, 6}}, {{0, 3}, {4, 9}}}*)

Another possibility using Outer:

Outer[Composition[Transpose, List], {v}, u, 1][[1]]

(* {{{0, 1}, {4, 3}}, {{0, 2}, {4, 6}}, {{0, 3}, {4, 9}}} *)

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I also like using Riffle for this:

Riffle[v, #] ~Partition~ 2 & /@ u