The nonlinear distortions resulting from the radio frequency (RF) impairments such as frequency offset, I/Q-imbalance, phase noise, and quantization errors. The hardware impairments severely degrade signal reception performance in NB-IoT systems and hence estimation for hardware impairments is necessary and mitigated by compensation algorithms. To this end, this paper focuses on the problem of estimating channel gains and frequency offsets in an NB-IoT system. Specifically, we first separate channel gains and frequency offsets and derive the analytical expressions of interest, based on the maximum likelihood estimation; We derive the expectation-maximization algorithm and apply it to the NB-IoT system for solving joint estimation of frequency offsets and channel gains; We carry out extensive simulations to illustrate that how systems parameters (e.g., channel phase difference, frequency offset difference, SNR, the number of sampling) can affect estimation performance.