Let H1 and H2 be complex Hilbert spaces. A bounded linear operator T:H1→H2 is called norm attaining if ‖Tx‖=‖T‖ for some unit vector x∈H1. If for every closed subspace M of H1, the restriction T|M:M→H2 is norm attaining, then T is called an absolutely norm attaining operator (or AN-operator). In the above definitions, if we replace the norm of the operator by the minimum modulus m(T)=inf⁡{‖Tx‖:x∈H1,‖x‖=1}, then T is called a minimum attaining and an absolutely minimum attaining operator (or AM-operator), respectively. In this article, we characterize Toeplitz AN-operators and discuss a few results on the minimum modulus of Toeplitz operator Tφ, φ∈L∞(T). We further characterize the minimum attaining Hankel operators and deduce that the only Hankel AM-operators are finite rank operators. While proving our results, we also obtained the following result; If φ∈L∞(T), then m(Lφ)=ess inf|φ| and there exists ψ∈L∞(T) such that γ(Lψ)>ess inf|ψ|, which improves a result from [15]. © 2022 Elsevier Inc.