Iwaniec, using the linear sieve, proved that $n^2+1$ can be a product of at most two primes infinitely often and furthermore a lower bound of the correct order of magnitude for the number of such integers $n$ is an expanded interval can be given.

My question is if there is a (possibly different) sieve that allows one to obtain an asymptotic in the case that a larger number of primes is allowed;i.e. letting $\omega$ stand for the number of prime factors and $$N_r(x):=\sharp\{n\leq x: \omega(n^2+1) \leq r\} $$ then is there a large but fixed value of $r$ such that one can prove an asymptotic for the quantity $N_r(x)$ ? I suspect that even correct upper bounds will be hard owing to parity issues but one can always hope.