Stimuli-responsive, readily functionalizable, and degradable nanogels have great potential in targeted drug delivery applications. While crosslinked polymeric nanoparticles such as nanogels modified with cell-targeting moieties can effectively deliver various hydrophobic drugs to cancer cells, they should be engineered to undergo degradation upon exposure to endogenous stimuli and release the payload once inside the cell. Herein, we report the preparation of a nanogel system crosslinked through the thiol–maleimide Michael addition reaction, which can be degraded in a reducing environment through a thiol–disulfide exchange reaction. The maleimide groups on the nanogels provided handles for conjugating cell-targeting motifs. To this end, a new monomer was designed to synthesize poly(ethylene glycol) methacrylate-based copolymers containing pendant maleimide groups linked to the polymer backbone through disulfide linkages. These copolymers were utilized to yield nanogels by crosslinking the nanoaggregates formed upon heating the copolymers in aqueous media. Nanogels were loaded with a clinically administered anticancer drug, docetaxel, and the drug release was investigated in the reductive cellular environment. Nanogels conjugated with cell-targeting peptides demonstrated preferential cellular internalization in breast cancer cells and also exhibited enhanced cytotoxicity in cells rich in glutathione. One can envision that such nanogels, employed to deliver various drugs and target motifs depending on the type of cancer, will be an attractive platform for such applications.