As stated in title I want to determine to which class
$$S=\left\{\langle M\rangle\Big\vert L(M)\in RE\setminus R\right\}$$
belongs. I believe that $S\notin RE\cup\text{co}RE$.
In order to show that $S\notin RE$ I tried to reduce $\overline{H_{TM}}$ to $S$ using the fact that $\overline{H_{TM}}\notin RE$
In order to show that $S\notin \text{co}RE$ I tried to reduce $H_{TM}$ to S using the fact that $H_{TM}\notin \text{co}RE$.
In both cases I got stuck with defining $M'$ and what it should do.
How should I construct the reduction? Thank you!