Let (Formula presented.) and (Formula presented.) be complex Hilbert spaces and (Formula presented.) be a bounded linear operator. We say (Formula presented.) is norm attaining if there exists (Formula presented.) with (Formula presented.) such that (Formula presented.). If for every non-zero closed subspace (Formula presented.) of (Formula presented.), the restriction (Formula presented.) is norm attaining, then (Formula presented.) is called an absolutely norm attaining operator or (Formula presented.) -operator. If we replace the norm of the operator by the minimum modulus (Formula presented.) in the above definitions, then (Formula presented.) is called a minimum attaining and an absolutely minimum attaining operator or (Formula presented.) -operator, respectively. In this article, we discuss the operator norm closure of (Formula presented.) -operators. We completely characterize operators in this closure and study several important properties. We mainly give a spectral characterization of positive operators in this class and give a representation when the operator is normal. Later, we also study the analogous properties for (Formula presented.) -operators and prove that the closure of (Formula presented.) -operators is the same as the closure of (Formula presented.) -operators. Consequently, we prove similar results for operators in the norm closure of (Formula presented.) -operators. © 2022 Informa UK Limited, trading as Taylor & Francis Group.