A strong clique in a graph G is a set S of edges of G such that every two edges in S are either adjacent or joined by an edge of G. We say that G is a circle graph if vertices of G correspond to chords of a circle and G contains an edge uv if and only if chords corresponding to u and v intersect.We show that the maximum size of a strong clique in a circle graph with maximum degree Δ is at most 54Δ2. This bound is tight for even Δ and correct up to a smaller order error term for odd Δ.This result supports a conjecture of Erdős and Nešetřil from 1985, which states that the strong chromatic index of a graph with maximum degree Δ is at most 54Δ2. It also confirms a conjecture of Faudree, Gyárfás, Schelp and Tuza from 1990 for circle graphs.