In this paper, we investigate the elements whose (b, c)-inverse is idempotent in a monoid. Let S be a monoid and a, b, c ∈ S. Firstly, we give several characterizations for the idempotency of a ||(b,c) as follows: a ||(b,c) exists and is idempotent if and only if cab = cb, cS = cbS, Sb = Scb if and only if both a ||(b,c) and 1 ||(b,c) exist and a ||(b,c) = 1 ||(b,c) , which establish the relationship between a ||(b,c) and 1 ||(b,c). They imply that a ||(b,c) merely depends on b, c but is independent of a when a ||(b,c) exists and is idempotent. Particularly, when b = c, more characterizations which ensure the idempotency of a ||b by inner and outer inverses are given. Finally, the relationship between a ||b and a ||b n for any n ∈ N + is revealed.