Context. Contemporary theory holds that massive stars gather mass during their initial phases via accreting disk-like structures. However, conclusive evidence for disks has remained elusive for most massive young objects. This is mainly due to significant observational challenges: objects are rare and located at great distances within dusty, highly opaque environments. Incisive studies, even targeting individual objects, are therefore relevant to the progression of the field. NGC 3603 IRS 9A* is a young massive stellar object that is still surrounded by an envelope of molecular gas for which previous mid-infrared observations with long-baseline interferometry have provided evidence of a plausible disk of 50 mas diameter at its core. Aims. This work aims at a comprehensive study of the physics and morphology of IRS 9A at near-infrared wavelengths. Methods. New sparse aperture-masking interferometry data, taken with the near-infrared camera NACO of the Very Large Telescope (VLT) at Ks and L′ wavelengths, were analyzed together with archival high-resolution H2 and Brγ lines obtained with the cryogenic high-resolution infrared schelle spectrograph (CRIRES). Results. The trends in the calibrated visibilities at Ks and L′-bands suggest the presence of a partially resolved compact object with an angular size of ≤30 mas at the core of IRS 9A, together with the presence of over-resolved flux. The spectroastrometric signal of the H2 line, obtained from the CRIRES spectra, shows that this spectral feature proceeds from the large-scale extended emission (~300 mas), while the Brγ line appears to be formed at the core of the object (~20 mas). To better understand the physics that drive IRS 9A, we have performed continuum radiative transfer modeling. Our best model supports the existence of a compact disk with an angular diameter of 20 mas, together with an outer envelope of 1′′ exhibiting a polar cavity with an opening angle of ~30°. This model reproduces the MIR morphology that has previously been derived in the literature and also matches the spectral energy distribution (SED) of the source. Conclusions. Our observations and modeling of IRS 9A support the presence of a disk at the core, surrounded by an envelope. This scenario is consistent with the brightness distribution (SED) of the source for near- and mid-infrared wavelengths at various spatial scales. However, our model suffers from remaining inconsistencies between SED modelling and the interferometric data. Moreover, the Brγ spectroastrometric signal indicates that the core of IRS 9A exhibits some form of complexity such as asymmetries in the disk. Future high-resolution observations are required to confirm the disk/envelope model and to flesh out the details of the physical form of the inner regions of IRS 9A.