Galaxy mergers occur frequently in the early Universe1 and bring multiple supermassive black holes (SMBHs) into the nucleus, where they may eventually coalesce. Identifying post-merger-scale (that is, less than around a few kpc) dual SMBHs is a critical pathway to understanding their dynamical evolution and successive mergers2. Whereas serendipitous discovery of ~kpc-scale dual SMBHs at z < 1 is possible3, such systems are elusive at z > 2 but critical in constraining the progenitors of SMBH mergers. The redshift z ≈ 2 also marks the epoch of peak activity of luminous quasars4, and therefore the probing of this spatial regime at high redshift is of particular importance in understanding the evolution of quasars. However, given stringent resolution requirements, there is currently no confirmed <10 kpc physical SMBH pair at z > 2 (refs. 5-8). Here, we report two sub-arcsec double quasars at z > 2 that were discovered from a targeted search with a novel astrometric technique, demonstrating a high success rate (≳50%) in this systematic approach. These high-redshift double quasars could be the long-sought kpc-scale dual SMBHs, or sub-arcsec gravitationally lensed quasar images. One of these double quasars (at z = 2.95) was spatially resolved with optical spectroscopy, and slightly favours the scenario of a physical quasar pair with a projected separation of 3.5 kpc (0.46″). Follow-up observations of double quasars discovered by this targeted approach will be able to provide observational constraints on kpc-scale dual SMBHs at z > 2.