The interaction between actin filaments and microtubules is crucial for many eukaryotic cellular processes, such as, among others, cell polarization, cell motility and cellular wound healing. The importance of this interaction has long been recognized, yet very little is understood about both the underlying mechanisms and the consequences for the spatial (re)organization of the cellular cytoskeleton. At the same time, understanding the causes and the consequences of the interaction between different biomolecular components are key questions for in vitro research involving reconstituted biomolecular systems, especially in the light of current interest in creating minimal synthetic cells. In this light, recent in vitro experiments have shown that the actin-microtubule interaction mediated by the cytolinker TipAct, which binds to actin lattice and microtubule tips, causes the directed transport of actin filaments. We develop an analytical theory of dynamically unstable microtubules, nucleated from the centre of a spherical cell, in interaction with actin filaments. We show that, depending on the balance between the diffusion of unbound actin filaments and propensity to bind microtubules, actin is either concentrated in the centre of the cell, where the density of microtubules is highest, or becomes localized to the cell cortex.