Recently, many methods have been proposed to generate a high spatial resolution (HR) hyperspectral image (HSI) by fusing HSI and multispectral image (MSI). Most methods need a precondition that HSI and MSI are well registered. However, in practice, it is hard to acquire registered HSI and MSI. In this paper, a synchronous nonnegative matrix factorization (SNMF) is proposed to directly fuse unregistered HSI and MSI. The proposed SNMF does not require the registration operation by modeling the abundances of unregistered HSI and MSI independently. Moreover, to exploit both HSI and MSI in the endmember optimization of the desired HR HSI, the unregistered HSI and MSI fusion is formulated as a bound-constrained optimization problem. A synchronous projected gradient method is proposed to solve this bound-constrained optimization problem. Experiments on both simulated and real data demonstrate that the proposed SNMF outperforms the state-of-the-art methods. © 2020, Springer Nature Switzerland AG.