The　numerical　manifold　method　(NMM)　is　a　partition　of　unity　(PU)　based　method.　For　the　purpose　of　obtaining　better　accuracy　with　the　same　mesh,　high　order　global　approximation　can　be　adopted　by　increasing　the　order　of　local　approximations　(LAs).　This,　however,　will　cause　the　＂linear　dependence＂　(LD)　issue,　where　the　global　matrix　is　rank　deficient　even　after　sufficient　constraints　are　enforced.　In　this　paper,　through　quadrilateral　mesh　to　form　the　mathematical　cover,　a　high　order　numerical　manifold　method　called　Quad4-COLS　(NMM)　is　developed,　where　the　constrained　and　orthonormalized　least-squares　method　(CO-LS)　is　used　to　construct　the　LAs.　The　developed　Quad4-COLS　(NMM)　does　not　need　extra　nodes　or　DOFs　to　construct　high　order　global　approximations,　while　is　free　from　the　LD　issue.　Nine　numerical　tests　including　five　tests　for　linear　elastic　continuous　problems　and　four　tests　for　linear　elastic　fracture　problems　are　carried　out　to　validate　the　accuracy　and　robustness　of　the　proposed　Quad4-COLS　(NMM).