Inspired by the discovery of a doubly charmed tetraquark state T + cc by the LHCb Collaboration and based on the prediction of a doubly charmed DD * molecule with I(J P ) = 0(1 + ) in our recent works, [Phys.Rev.D 102 (2020), 091502] and [Phys.Rev.D 99 (2019), 094018], we employ the effective Lagrangian approach to investigate the decay width of T cc → DDπ and T cc → DDγ. We show that both the T cc → DDπ and T cc → DDγ modes contribute to the decay width of T cc , with the former playing a more important role. However, within the DD * molecule picture, the obtained decay width is rather small compared with the experimental value of Γ = 410 ± 175 keV. We argue that the existence of a compact tetraquark component cannot lead to an appreciable increase of the decay width. As a result, the discrepancy may indicate that either T cc is more likely a near threshold DD * resonance or the decay width is close to the lower experimental boundary.