Is this language recursively enumerable?
Thanks to @Yuval and @chi, I've been able to find a working reduction.

The reduction is $\overline{L_{halt}} \leq L$. For an input $&lt;M&gt;&lt;w&gt;$ we build the following TM $N_w$:


Runs $M$ on $w$ for $k$ steps, where $k$ is the length of the input for $N_w$.
If $M$ was able to accept or reject $w$ within $k$ steps in the above simulation, $N_w$ will reject. Otherwise, accept.


This way, if $&lt;M&gt;&lt;w&gt;\in \overline{L_{halt}}$ then  $M$ does not halt on $w$ for any length $k$, thus $N_w$ will accept every string. Then, $L(N_w)=\Sigma^*\in P$ and $&lt;N_w&gt;\in L$. 

Else, if $&lt;M&gt;&lt;w&gt;\notin \overline{L_{halt}}$ then a $n\in\mathbb{N}$ exists s.t. $M$ halts on $w$ within $n$ steps. Thus, for all inputs of length $\geq n$ the TM $N_w$ will reject, and it will only accept inputs of length $ &lt; n$, and not satisfaying the property $P$. Hence, $&lt;N_w&gt;\notin L$.

And since $\overline{L_{halt}}\notin RE$, it follows that $L\notin RE$ as well.