An oriented closed, connected \(n\)-manifold \(M\) is inflexible if the self-maps of \(M\) are of bounded degree. The manifold \(M\) is strongly inflexible if all maps from any other oriented closed, connected \(n\)-manifold to \(M\) have bounded degree. Inflexible manifolds arise in the construction of functorial semi-norms on manifolds [M. Gromov, Metric structures for Riemannian and non-Riemannian spaces. Transl. from the French by Sean Michael Bates. With appendices by M. Katz, P. Pansu, and S. Semmes. Edited by J. LaFontaine and P. Pansu. Boston, MA: Birkhäuser (1999; Zbl 0953.53002)]. Examples of simply connected inflexible manifolds have been constructed using Sullivan minimal models, beginning with an example in [M. Arkowitz and G. Lupton, Math. Z. 235, No. 3, 525–539 (2000; Zbl 0968.55005)], and including the solutions to the realization problem for finite groups as groups of self-homotopy equivalences given in [C. Costoya and A. Viruel, Acta Math. 213, No. 1, 49–62 (2014; Zbl 1308.55005)]. On the other hand, there are no known examples of strongly inflexible manifolds. In this paper, the authors give a Sullivan model criterion that implies a manifold \(M\) is not strongly inflexible. Let \((\land V, d)\) be Sullivan the minimal model for \(M\) with fundamental class represented by a cycle \(\eta \in (\land V)^n\). Suppose there is a dg algebra map \(\psi \colon (\land V, d) \to (\mathcal{A}, d)\) such that \((\mathcal{A}, d)\) has positive weight and \(H^n(\psi)(\eta) \neq 0.\) Then \(M\) is not strongly flexible. The result is applied to prove all the known examples of inflexible manifolds, except one whose status remains an open problem, are not strongly inflexible.