For systems with constraints the issue of non-commutability of quantization and elimination of non-physical variables is studied in the framework of path integrals (PIs). It is shown that one should take into consideration the curvilinearity of physical variables and their phase space reduction in order to provide a one-to-one correspondence between the Dirac scheme and PI description. The latter leads to a modification of the standard PI (PI with a gauge condition). A general PI derivation is suggested for any method of picking out physical variables which corresponds to the Dirac scheme.