We investigate the problem of uncoordinated massive random access in the quasi-static asynchronous Rayleigh fading channel. In the previous work , the authors assumed a completely synchronous scenario which is impossible in any practical implementation. This paper extends the previous work to the asynchronous case. As energy efficiency is of critical importance for massive machine-type communication (mMTC), our main goal is to minimize the energy-per-bit required to achieve the target probability of error. Another issue required for mMTC is a transmitter simplicity. As in the synchronous case, we focus on grant-free transmission and do not use preambles and other synchronization sequences. We propose a practical implementation of a transmission scheme based on synchronization error estimation and cancellation in the frequency domain. The simulation shows that the proposed transmission scheme's performance is very close to the synchronous case. The only source of E_b/N_0 loss is the need for an additional cyclic prefix that helps to solve the synchronization error cancellation problem in the frequency domain.