The recently discovered heavy-fermion superconductor, UTe2, is an excellent candidate for spin-triplet superconductors in which electrons form spin-triplet Cooper pairs with spin S = 1 and odd parity. Unconventional superconductivity often hosts unconventional vortices. Yet, the vortex core and lattice in UTe2 have not been directly visualized and characterized. Here, by using ultralow-temperature scanning tunnelling microscopy and spectroscopy, we study the superconducting vortices on the (0−11) surface termination of UTe2 with an out-of-plane external magnetic field. At the centre of the vortex core, we observe a robust zero-energy vortex-core state that exhibits a cigar-shaped spatial distribution and extends to ∼30 nm along the [100] direction (crystallographic a-axis) of UTe2. Along the direction perpendicular to [100], the superconducting gap is deeper and the coherence peak on one side of the vortex core is stronger than on the opposite side, and they are even enhanced in comparison with those under zero field. Due to the anisotropy of magnetic susceptibility in UTe2, the asymmetric dI/dV spectra on the two sides of the vortex core result from the interplay between the magnetization-induced bound current and supercurrent around the vortex core. Our work reveals the important role of magnetization in the vortex behaviours of UTe2 and provides essential microscopic information for understanding its superconducting properties in a magnetic field.