Given a complete \(m\)-dimensional Riemannian manifold \((M,g_0)\) with a pole \(x_0,\) define \(r(x)=\text{dist}_{g_0}(x,x_0).\) Let \((N,h)\) be an \(n\)-dimensional Riemannian manifold. Assuming that \(f\colon\;(M,\eta^2g_0)\to (N,h)\) is an exponentially harmonic map with a smooth function \(\eta\colon\;M\to \mathbb{R},\) satisfying \(\frac{\partial \log\eta}{\partial r}\geq0,\) the author derives certain monotonicity formulae for the exponential energy functional.