We consider a random process as a solution of stochastic differential equations with dependence of the coefficients on small parameter $\varepsilon$ and we suppose that the drift coefficients of these equations are unbounded on the parameter $\varepsilon$. We consider more general requirements on the convergence of some functions of coefficients of stochastic equations to limit functions. Necessary and sufficient conditions for the weak convergence of solutions of such stochastic equations if $\varepsilon$ tends to zero to a some stochastic equations involving a local time of process are obtained.