Abstract We study phenomenologically the density dependence of the K − optical potential V opt( r) at zero kinetic energy by fitting it to a comprehensive set of atomic data across the periodic table. Two families of solutions offer improved fits over that for the standard t eff ϱ( r) paraameterization of V opt. One family contains solutions for V opt with RMS radii larger than those of the matter density ϱ, as expected from finite-range folding corrections. The other family offers substantially improved fits to the data and gives rise to Re V opt of a size smaller than that of ϱ. The new family contains solutions for which V opt is attractive inside the nucleus and on its surface, becoming repulsive at large distances. The repulsion can be made to approximately agree with the low-density limit V opt→ tϱ where t is the free-space KN t matrix. It is argued that this sign flip might be expected from the underlying nuclear propagation of the Λ(1405).