In this paper the authors classify, for given integers \(m,n\geq 1\), the bordism class of a closed \(m\)-dimensional smooth manifold \(X^m\) with a free smooth involution \(\tau\) with respect to the validity of the Borsuk-Ulam property (BUP) that for every continuous map \(\phi:X\to \mathbb{R}^n\) there exists a point \(x\in X^n\) such that \(\phi(x)=\phi(\tau(x))\). They classify for each \(m\)-dimensional free \(\mathbb Z_2\)-bordism class \(\alpha\) and integer \(n\), whether the BUP with respect to \(\mathbb{R}^n\) holds (a) for all of the representatives \((X^m,\tau)\) of \(\alpha\), or (b) for some, but not all, of the representatives, or (c) for none them.