We construct exchange energy functionals that depend on the non-interacting square kinetic energy density tau(2)(r)=1/4<&USigma;(k)\&UDelta;φ(k)(r)\(2) rather than the electron density. Since τ(2) is a non-local functional of the electron density, it may describe non-local effects beyond standard generalized gradient approximation models. These effects may be essential in dealing with such a non-local quantity as the exchange energy. The problem of v-representability of τ(2) for a slightly perturbed electron gas and Coulomb systems is discussed. A gradient expansion technique that permits construction of gradient-corrected functionals of τ(2) is developed, and the performance of the new exchange functionals is analyzed.