We present the draft genome sequences of two clinical strains of Aspergillus turcosus, one azole-susceptible (strain HMR-AF-23/LSPQ-01275) and the other azole-resistant (strain HMR-AF-1038/LSPQ-01280), isolated from bronchoalveolar lavage fluid of two adult patients. These two strains are the first reported clinical isolates of A. turcosus.