V. Istratescu has recently defined $M$-paranormal operators on a Hilbert space $H$ as: An operator $T$ is called $M$-paranormal if for all $x\in H$ with $\|x\|=1$, $$ \|T^2 x\|\geqq\frac1M\|Tx\|^2 $$ We prove the following results: \item{1.} $T$ is $M$-paranormal if and only if $M^2T^*2T^2-2\lambda T^*T+\lambda^2 \geq 0$ for all $\lambda > 0$. \item{2.} If a $M$-paranormal operator $T$ double commutes with a hyponormal operator $S$, then the product $TS$ is $M$-paranormal. \item{3.} If a paranormal operator $T$ doble commutes with a $M$-hyponormal operator, then the product $TS$ is $M$-paranormal. \item{4.} If $T$ is invertible $M$-paranormal, then $T^{-1}$ is also $M$-paranormal. \item{5.} If $Re W (T) \leq 0$, where $W (T)$ denotes the numerical range of $T$, then $T$ is $M$-paranormal for $M \geq 8$. \item{6.} If a $M$-paranormal partial isometry $T$ satisfies $\|T\| \leq \frac1M$, then it is subnormal.