Nitrogen adsorption on a Cu(001) surface leads to formation of the grid-like nanopattern where square c(2 × 2) N-adsorbed islands of the size, 5nm×5nm, are aligned along [100] and [010] directions. The grid pattern leaves bare Cu(001) regions (“patches”) at grid intersections. Using scanning tunneling microscopy, we find that short bright lines running in [110] directions appear on patch edges only when the grid pattern approaches completion. We propose the structural model for the bright line that a Cu row moves by half period to form local (111)-like close-packed structure. The local structure can relieve effectively compressive stress imposed by expanding N-adsorbed islands. The model can be smoothly connected with our missing-Cu-row model proposed for a narrow boundary (“monoatomic line”) between N-adsorbed islands [M. Yamada et al., Surf. Sci. 604 (2010) 1961]. Unlike heavy 5d metals such as Au, a Cu(001) surface keeps its bulk-truncated structure. However, it is shown that large compressive stress can induce local (111)-like reconstruction on Cu(001).