We study the holomorphic vector bundles E over the twistor space \({{\,\mathrm{Tw}\,}}(M)\) of a compact simply connected hyperkähler manifold M. We give a characterization of the semistability condition for E in terms of its restrictions to the holomorphic sections of the holomorphic twistor projection \(\pi \,:\, {{\,\mathrm{Tw}\,}}(M)\,\longrightarrow \, {\mathbb {CP}}^1\). It is shown that if E admits a holomorphic connection, then E is holomorphically trivial and the holomorphic connection on E is trivial as well. For any irreducible vector bundle E on \({{\,\mathrm{Tw}\,}}(M)\) of prime rank, we prove that its restriction to the generic fibre of \(\pi \) is stable. On the other hand, for a K3 surface M, we construct examples of irreducible vector bundles of any composite rank on \({{\,\mathrm{Tw}\,}}(M)\) whose restriction to every fibre of \(\pi \) is non-stable. We have obtained a new method of constructing irreducible vector bundles on hyperkähler twistor spaces; this method is employed in constructing these examples.