Find negative cycle with vertex constraints
If you don't require the cycle to be simple, then break the (directed) graph into its strongly connected components, and for each component containing one of the given vertices $V_i$, check whether the component contains a negative cycle.  If no component does, there is no negative cycle containing any $V_i$.  But if some component does, you can find a (non-simple) negative cycle containing $V_i$ by taking many copies of the negative cycle, and adding to that paths to and from some vertex in the cycle to $V_i$.  (The total time to find an implicit representation of the desired cycle will be the same as the time to find a negative cycle in a directed graph, e.g. $O(nm)$, if I recall.)

If you do require the cycle to be simple, then the problem becomes NP-complete, even if only a single vertex $V_1$ is given.  (You can reduce Hamiltonian Path to the problem: to find a Hamiltonian path from a given source $S$ to a given sink $T$ in a given graph $G$, give the existing edges weight -1, then add an artificial vertex $V_1$ with two edges of cost $N/2-0.01$ each, one from $V_1$ to $S$ and one from $T$ to $V_1$.)

If you allow the cycle to repeat vertices but not edges, I believe it is still NP-complete (by a similar reduction, but splitting each vertex $v$ into a directed edge $(v,v')$ in a standard way).