We first show that already a moderate allowed negation width can substantially decrease the size of monotone circuits. Our main result is a general reduction from non-monotone circuits of bounded negation width to monotone circuits: if a monotone Boolean function $f$ can be computed by a non-monotone circuit of size $s$ and negation width $w$, then $f$ can be also computed by a monotone circuit of size $s$ times $4\min\{w^m,m^w\}\log M$, where $m$ is the maximum length of a prime implicant and $M$ is the total number of prime implicants of~$f$.