Magnetic core–shell nanoparticle (NP)-polymer-based composite materials, in which the stimuli-responsive polymer acts as a suitable shell, become more advantageous as nanocarriers. Here, nanocomposites comprising magneto-responsive γ-Fe2O3 NPs that incorporated amphiphilic copolymers have been synthesized. At first, the copolymer [dextran grafted with poly(N-vinyl caprolactam)] has been prepared via reversible addition–fragmentation chain transfer polymerization followed by ex situ incorporation of γ-Fe2O3 NPs. The copolymerization is living in nature as apparent from molecular weight distribution (determined using advanced polymer chromatography, i.e., APC analysis) and the reaction kinetics study. The developed nanocomposite exhibits regular core–shell morphology, where the magnetic NPs have been found to behave as the core, while the copolymer represents as the shell, as obvious from field emission scanning electron microscopy and high-resolution transmission electron microscopy analyses. The nanocarrier is non-cytotoxic and has further been studied for its effectivity as an efficient carrier for a hydrophobic drug (naproxen).