In this work, we consider a Riemannian manifold $$M$$ with an almost quaternionic structure $$V$$ defined by a three-dimensional subbundle of $$(1,1)$$ tensors $$F$$ , $$G$$ , and $$H$$ such that $$\{F,G,H\}$$ is chosen to be a local basis for $$V$$ . For such a manifold there exits a subbundle $$\mathcal{{H}} (M)$$ of the bundle of orthonormal frames $$\mathcal{{O}}(M)$$ . If $$M$$ admits a torsion-free connection reducible to a connection in $$\mathcal{{H}}(M)$$ , then we give a condition such that the torsion tensor of the bundle vanishes. We also prove that if $$M$$ admits a torsion-free connection reducible to a connection in $$\mathcal{{H}}(M)$$ , then the tensors $$\widetilde{F}^2$$ , $$\widetilde{G}^2$$ , and $$\widetilde{H}^2$$ are torsion-free, that is, they are integrable. Here $$\widetilde{F}$$ , $$\widetilde{G}$$ , $$\widetilde{H}$$ are the extended tensors of $$F$$ , $$G$$ , and $$H$$ defined on $$M$$ . Finally, we show that if the torsions of $$\widetilde{F}^2$$ , $$\widetilde{G}^2$$ and $$\widetilde{H}^2$$ vanish, then $$M$$ admits a connection with torsion which is reducible to $$\mathcal{{H}}(M)$$ , and this means that $$\widetilde{F}^2$$ , $$\widetilde{G}^2$$ , and $$\widetilde{H}^2$$ are integrable.