For a precise characterization of time-domain fluorescence lifetime imaging microscopy (FLIM) datasets, an initial processing step is needed to identify the fluorescent impulse response (FIR) at each spatial point in the sample. Hence departing from the measured fluorescent decays, the FIRs are estimated by using the instrument response function (IRF), and this processing step is known as deconvolution. However, the deconvolution methodology requires an initial measurement of the IRF and a corresponding synchronization step with the fluorescent decays. In this context, we propose a blind deconvolution strategy that estimates jointly the FIRs and the IRF in the dataset. For this purpose, each FIR is modeled by a multi-exponential structure. In this way, the FIRs are characterized by the scaling coefficients and time constants of the exponential terms. Meanwhile, there is no explicit model or pre-defined shape for the IRF. Overall estimation process is achieved by an alternated least squares methodology between the FIRs and IRF. First, if the IRF is fixed, a nonlinear least squares framework computes the FIRs parameters at each spatial point of the sample. Meanwhile, once the FIRs are fixed, the samples of the IRF are estimated by a non-negative least squares methodology and using the whole dataset. These alternated optimization steps are performed until a convergence criterion is fulfilled. The proposed blind deconvolution strategy was validated by synthetic datasets and in vivo FLIM oral mucosa measurements. In these tests, our proposal shows good characterizations of the FIRs and the IRFs in the FLIM datasets.