A series of amphiphilic diblock copolymers with different hydrophilic and hydrophobic block lengths were synthesized by reductive amination of hydroxypropyl methyl cellulose (HPMC) and end-functionalized poly(ε-caprolactone) (PCL). NH2-PCL was synthesized by ring-opening polymerization of ε-caprolactone using 3-(Boc-amino)-1-propanol as the initiator, followed by removal of the Boc group with trifluoroacetic acid treatment. Proton NMR, DOSY-NMR, Fourier transform infrared, and the Kaiser test confirmed the successful synthesis of HPMC–PCL diblock copolymers. The copolymers were able to self-assemble by dissolution in water, yielding spherical nanoscale micelles of ca. 200 nm, as revealed by transmission electron microscopy and dynamic light scattering. The micelle size was mainly dependent on the hydrophilic HPMC block length. The longer the HPMC block, the larger the micelle size. The critical micelle concentration determined by fluorescence spectroscopy decreased with the increasing hydrophobic PCL block length. Curcumin was loaded in copolymer micelles as a lipophilic antitumor drug. Higher drug loading was obtained for copolymers with longer PCL blocks. The in vitro drug release profile consisted two phases with an initial burst and a subsequent slower release. Faster drug release is observed for copolymers with a shorter PCL block length and lower drug loading. HPMC–PCL copolymer micelles exhibited outstanding cytocompatibility, as evidenced by the MTT test using L929 cells. Therefore, it is concluded that biodegradable and biocompatible HPMC–PCL block copolymers present great potential as nanocarriers of lipophilic antitumor drugs.