Local cohomology modules and direct limits

Question

Let $(R, \frak{m})$ be a Noetherian local ring and $M$ be an $R$-module such that $H^i_{\frak m} (M)=0$ for all $i>0$ ($H^i_{\frak{m}}(M)$ denotes the $i$-th local cohomology module of $M$ with respect to $\frak{m}$). We know that $M$ is a direct limit of finitely generated $R$-modules say $\{ M_i \}_{i \in I}$. Can we find this directed system such that $H^i_{\frak m}(M_j)=0$ for all $i>0$ and all $j \in I$?