We have $$ V^{pv}(k)=-\frac{f_{\pi}^{2}}{m_{\pi}^{2}} \frac{(\sigma_{1}\cdot\mathbf{k})(\sigma_{2}\cdot\mathbf{k})}{\mathbf{k}^{2}+m_{\pi}^{2}}. $$ In coordinate space we have $$ V^{pv}(r)=\int\frac{d^3k}{(2\pi)^3}e^{i\mathbf{kr}}V^{pv}(k) $$ resulting in $$ V^{pv}(r)=-\frac{f_{\pi}^{2}}{m_{\pi}^{2}} \sigma_{1}\cdot{\nabla}\sigma_{2}\cdot{\nabla} \int\frac{d^3k}{(2\pi)^3}e^{i\mathbf{kr}}\frac{1}{\mathbf{k}^{2}+m_{\pi}^{2}}. $$