The authors consider a problem of local controllability for a nonlinear control system of the form \((1)\quad x'(t)=f_ 0[x(t)]+\sum^{m}_{i=1}f_ i[x(t)]u_ i(t),\quad x(0)=p\in {\mathbb{R}}^ n;\quad | u_ i(t)| \leq 1,\quad i=1,...,m\) where \(x\in {\mathbb{R}}^ n\), \(f_ i\) are analytical vector-functions, \(u_ j\) are measurable control functions. Let G(p;0,t) be the attainable set of the system (1) at time t and \(G(p;[0,t])=\cup_{s\in [0,t\}}G(p;0,s).\) The system (1) is said to have a p-local attainability property on [0,t] if \(p\in int_{{\mathbb{R}}^ n}G(p;[0,t])\). The authors give necessary conditions and sufficient conditions for the p-local attainability for the system (1) generalizing some known results.