  We prove the first rigorous bound on the heat transfer for three-dimensional
Rayleigh-B\'enard convection of finite-Prandtl-number fluids between free-slip
boundaries with an imposed heat flux. Using the auxiliary functional method
with a quadratic functional, which is equivalent to the background method, we
prove that the Nusselt number $Nu$ is bounded by $Nu \leq 0.5999 R^{1/3}$
uniformly in the Prandtl number, where $R$ is the Rayleigh number based on the
imposed heat flux. In terms of the Rayleigh number based on the mean vertical
temperature drop, $Ra$, we obtain $Nu \leq 0.4646 Ra^{1/2}$. The scaling with
Rayleigh number is the same as that of bounds obtained with no-slip isothermal,
free-slip isothermal, and no-slip fixed flux boundaries, and numerical
optimization of the bound suggests that it cannot be improved within our
bounding framework. Contrary to the two-dimensional case, therefore, the
$Ra$-dependence of rigorous upper bounds on the heat transfer obtained with the
background method for three-dimensional Rayleigh-B\'enard convection is
insensitive to both the thermal and the velocity boundary conditions.
