Metamaterials　have　been　extensively　investigated　owing　to　their　unusual　properties.　However,　the　design　of　multifunctional　metamaterials　requires　further　investigation.　This　study　focused　on　the　design　of　three-dimensional　(3D)　metamaterials　to　achieve　tunable　negative　thermal　expansion　and　phononic　bandgap　properties.　First,　the　independent　continuous　mapping　(ICM)　topology　optimisation　method　was　applied　to　create　metamaterial　microstructures　with　negative　thermal　expansion　properties　based　on　the　multi-scale　asymptotic　homogenisation　theory.　Secondly,　the　conceptual　structure　from　the　topology　optimisation　was　reconstructed　and　parameterized　to　achieve　the　desired　phononic　bandgap　widths　under　negative　thermal　expansion,　using　a　surrogate　model-based　optimisation　method.　Both　the　negative　coefficient　of　thermal　expansion　and　phononic　bandgaps　were　verified　through　numerical　simulations.　The　results　reveal　that,　by　selecting　appropriate　parameters,　the　designed　metamaterials　can　have　both　a　negative　coefficient　of　thermal　expansion　and　a　maximum　bandgap　width　ratio.　The　proposed　method　provides　an　important　reference　for　the　rational　design　of　multifunctional　metamaterials.　CO　2021　The　Authors.　Published　by　Elsevier　Ltd.　This　is　an　open　access　article　under　the　CC　BY-NC-ND　license　(http://creativecommons.org/licenses/by-nc-nd/4.0/).