Stimuli-responsive micellar nanoparticles are gaining considerable attention in the field of drug delivery because of their great advantages of high efficiency in tumor accumulation through the enhanced permeability and retention effect and rapid drug release. In this study, a novel catechol moiety-containing amphiphilic, biodegradable polymeric carrier, monomethoxy poly(ethylene glycol)-block-poly(ε-caprolactone)-block-poly(L-lysine)-graft-[3-(3,4-dihydroxyphenyl)propionic acid], was employed for delivery of the anticancer drug bortezomib (BTZ) to cancer cells. The strategy is based on the facile conjugation of BTZ to catechol-containing polymeric carriers through boronate formation, which could dissociate in acidic environments to liberate its payload BTZ to inhibit proteasome function. This reducing BTZ–catechol interaction with decreasing pH value was demonstrated by microcalorimetry analysis using isothermal titration calorimetry. Notably, the obtained micellar BTZ complex that self-assembled from drug-loaded amphiphilic polymer was internalized effectively by MCF-7 breast cancer cells and resulted in significant 26S proteasome inhibition. Furthermore, the micellar BTZ complex induced remarkable apoptosis on MCF-7 and HeLa cancer cells and exhibited little toxicity on HEK293 normal cells. More importantly, systemic delivery of micellar BTZ complex prolonged its blood circulation and resulted in significant accumulation in the tumor site relative to free drug, thus suggesting therapeutic promise for micellar BTZ delivery in cancer therapy.