We propose a hybrid quantum system consisting of a magnetic particle inductively coupled to two superconducting transmon qubits, where qubit-qubit interactions are mediated via magnons. We show that the system can be tuned into three different regimes of effective qubit-qubit interactions, namely, a transverse (XX+YY), a longitudinal (ZZ), and a nontrivial ZX interaction. In addition, we show that an enhanced coupling can be achieved by employing an ellipsoidal magnet, carrying anisotropic magnetic fluctuations. We propose a scheme for realizing two-qubit gates, and simulate their performance under realistic experimental conditions. We find that iswap and cz gates can be performed in this setup with an average fidelity ≤99%, while an icnot gate can be applied with an average fidelity ≤88%. Our proposed hybrid circuit architecture offers an alternative platform for realizing two-qubit gates between superconducting qubits and could be employed for constructing qubit networks using magnons as mediators.