Summary: It is shown that given any finite system of monoid identities, it is decidable if the class of locally finite monoids that satisfy the system is a variety. This answers an open problem of M. V. Sapir [Combinatorial algebra: syntax and semantics. With contributions by Victor S. Guba and Mikhail V. Volkov. Cham: Springer (2014; Zbl 1319.05001)].