Given any reduced excellent scheme $X$, there exists a *Macaulayfication* $\pi : Y \to X$. In other words, there exists a proper birational map $\pi$ from a Cohen-Macaulay scheme $Y$ to $X$. These exist by a result of Kawasaki.

Is it possible that we may also pick $Y$ to be normal? I see no reason why this should be true from the construction (blowing up various systems of parameters). Of course, when resolutions of singularities exist, this certainly solves the problem...

One might hope that the normalization of a Cohen-Macaulay scheme is Cohen-Macaulay, but I believe this is false (for example given a normal non-CM variety in characteristic zero, you should be able to generically project it to a hypersurface).