We investigate the relationship between the Levitin--Polyak well-posedness of the problem of minimization of the integral functional $ I:x\in L^1(T)\to \int_Tf(t,x(t)) dt $ on the set $U=\left\{ x:T\subset R^k\to R^m: x\in L^1(T);\ x(t)\in K(t)\text{ for a.e. }t\in T\right\}$ of integrable selections of a multifunction $K:t\in T\to K(t)\subset R^m$ and well-posedness of the minimization problem of $f(t,.)$ on $K(t)$. We show that well-posedness of problem $\inf(I,U)$ implies that of $\inf(f(t,.),K(t))$ for a.e. $t\in T$. The converse holds under another assumptions.