In this paper, we first study infinitesimal deformations of a Lie conformal algebra and a Lie conformal algebra with a module (called an LCMod pair), which lead to the notions of Nijenhuis operator on the Lie conformal algebra and Nijenhuis structure on the LCMod pair, respectively. Then by adding compatibility conditions between Nijenhuis structures and O-operators, we introduce the notion of an ON-structure on an LCMod pair and show that an ON-structure gives rise to a hierarchy of pairwise compatible O-operators. In particular, we show that compatible O-operators on a Lie conformal algebra can be characterized by Nijenhuis operators on Lie conformal algebras. Finally, we introduce the notions of conformal r-matrix-Nijenhuis structure and symplectic-Nijenhuis structure on the Lie conformal algebra and study their relations. Contents 1. Introduction 1 2. Preliminaries on Lie conformal algebra 3 3. Nijenhuis operators and Nijenhuis structures on Lie conformal algebras 6 3.1. Nijenhuis operators on Lie conformal algebras 6 3.2. Nijenhuis structures on LCMod pairs 11 4. ON-structures and compatible O-operators on LCMod pairs 13 5. Conformal r-matrix-Nijenhuis structures and symplectic-Nijenhuis structures on Lie conformal algebras 19 5.1. Conformal r-matrix-Nijenhuis structures on Lie conformal algebras 20 5.2. Symplectic-Nijenhuis structures on Lie conformal algebras 21 References 25 2010 Mathematics Subject Classification. 17A30, 17B69, 17B62.