Combining a high-spatial-resolution multispectral image (HR-MSI) with a low-spatial-resolution hyperspectral image (LR-HSI) has become a common way to enhance the spatial resolution of the HSI. The existing state-of-the-art LR-HSI and HR-MSI fusion methods are mostly based on the matrix factorization, where the matrix data representation may be hard to fully make use of the inherent structures of 3-D HSI. We propose a nonlocal sparse tensor factorization approach, called the NLSTF_SMBF, for the semiblind fusion of HSI and MSI. The proposed method decomposes the HSI into smaller full-band patches (FBPs), which, in turn, are factored as dictionaries of the three HSI modes and a sparse core tensor. This decomposition allows to solve the fusion problem as estimating a sparse core tensor and three dictionaries for each FBP. Similar FBPs are clustered together, and they are assumed to share the same dictionaries to make use of the nonlocal self-similarities of the HSI. For each group, we learn the dictionaries from the observed HR-MSI and LR-HSI. The corresponding sparse core tensor of each FBP is computed via tensor sparse coding. Two distinctive features of NLSTF_SMBF are that: 1) it is blind with respect to the point spread function (PSF) of the hyperspectral sensor and 2) it copes with spatially variant PSFs. The experimental results provide the evidence of the advantages of the NLSTF_SMBF method over the existing state-of-the-art methods, namely, in semiblind scenarios.