In this paper we study $n$-dimensional compact minimal submanifolds in $S^{n+p}$ with scalar curvature $S$ satisfying the pinching condition $S>n(n-2)$. We show that for $p\leq 2$ these submanifolds are totally geodesic (cf. Theorem 3.2 and Corollary 3.1). However, for codimension $p\geq 2$, we prove the result under an additional restrictions on the curvature tensor corresponding to the normal connection (cf. Theorem 3.1 and Corollary 4.1). We also show that the scalar curvature $S$ of a non-totally geodesic $n$-dimensional non-negatively curved minimal submanifold in $S^{n+p}$ with flat normal connection satisfies $n(n-p-1)\leq S\leq n(n-2)$ (cf. Theorem 4.1). Since for a compact hypersurface $M$ of $S^{n+1}$ the normal connection is flat, we use the above estimate for a scalar curvature $S$ of a non-negatively curved minimal hypersurface $M$ in $S^{n+1}$ to infer that either $M$ is totally geodesic or else it is isometric to the hypersurface $S^m\left(\sqrt{\frac{m}{n}}\right)\times S^{n-m}\left(\sqrt{ \frac{n-m}{n}}\right)$. As a consequence this result, we conclude that the only non-negatively curved compact minimal hypersurfaces in $S^{n+1}$ which are diffeomorphic to $S^n$ is totally geodesic sphere.