Not that I'm aware of.
If you look at the real space representation of the computations, they're based on the Green's function of the Klein-Gordon equation (and its first derivative, if you're using fermions) which is nice and confined to within the light cones (forward and backward). Usually, we use the Feynman propagator, which includes the Green's function in its real part and the vacuum fluctuations in its imaginary part. The vacuum fluctuations have correlations outside of the light cone, but those aren't propagating signals, they're coincidences enforced by the fact that the spectrum of ground state fluctuations is fixed by the Hamiltonian. The formula for the Feynman propagator is
\begin{align}
\Delta_F(s) = -\frac{1}{4\pi} \delta(s) - \frac{imc}{4\pi^2 \hbar\sqrt{s}} K_1\left(\frac{mc\sqrt{s}}{\hbar}\right),
\end{align}
where $K_1$ is a modified Bessel function of the second kind, $m$ is the mass of the particle, and if the propagator is between the spacetime points $x$ and $y$ then $s = (\mathbf{x}-\mathbf{y})^2 - (x^0 - y^0)^2$. The important thing is that the imaginary part of $\Delta_F$ comes from vacuum correlations, and the real part is what deals with the creation and annihilation of virtual particles.
The idea that virtual particles can travel faster than light comes from looking at the Fourier space representation of the Feynman propagator. If we take the full 4-$d$ Fourier transform of $\Delta_F$ using the unitary convention we get that
\begin{align}
\Delta_F(p) &=\frac{\hbar^2}{\left(2\pi\right)^2\left[-(p^0)^2 c^{-2} + \mathbf{p}^2 + m^2c^2\right]} \nonumber \\
&\hphantom{=}+ \frac{2\pi i}{(2\pi)^2} \delta\left(\frac{-(p^0)^2 c^{-2} + \mathbf{p}^2 + m^2c^2}{\hbar^2}\right).
\end{align}
Note that the usual formulation only includes the first term, and the second term is produced by a choice of how to dodge the pole in the propagator using a complex contour integral. In the above formulation, the pole is assumed to be integrated using the Cauchy principle value, and the mass shell term is included explicitly. Because there is now no contour integral, you can see that the part of the propagator that is confined to the mass shell (the dispersion relation) produces the real-space vacuum fluctuations. Those Fourier modes travel at $c$ (or less) because they obey a dispersion relation
\begin{align}
\omega^2 = \frac{m^2 c^4}{\hbar^2} + \mathbf{k}^2c^2.
\end{align}
Does that mean that the propagator requires particles that travel faster than light to get the real part of the real space propagator? Not really. There is a formalism known as "on shell recursion." I haven't studied it in detail, so I cannot comment on how they do it, but the simple implication is that they don't use off-shell virtual particles. What I can comment on is that the on-shell part of the Feynman propagator is $p^0$-even. The $p^0$-odd version has the Fourier space form
\begin{align}
\frac{2\pi i}{(2\pi)^2} \operatorname{sgn}(p^0) \ \delta\left(\frac{-(p^0)^2 c^{-2} + \mathbf{p}^2 + m^2c^2}{\hbar^2}\right).
\end{align}
If we Fourier transform that back to real space, we get
\begin{align}
\operatorname{sgn}(t)\,\Theta([ct]^2 - r^2) \frac{mc J_1\left(\frac{mc\sqrt{(ct)^2 - r^2}}{\hbar}\right)}{4\pi \hbar \sqrt{(ct)^2 - r^2}}.
\end{align}
Note that, up to the sign function, that's identical to the real part of the Feynman propagator. In principle, then, it should be possible to work with only on mass shell modes in the Fourier transform of the propagator, if you're careful enough.
