  Let $p\geq 5$ be a prime number. We consider the Iwasawa $\lambda$-invariants
associated to modular Bloch-Kato Selmer groups, considered over the cyclotomic
$\mathbb{Z}_p$-extension of $\mathbb{Q}$. Let $g$ be a $p$-ordinary cuspidal
newform of weight $2$ and trivial nebentype. We assume that the $\mu$-invariant
of $g$ vanishes, and that the image of the residual representation associated
to $g$ is suitably large. We show that for any number greater $n$ greater than
or equal to the $\lambda$-invariant of $g$, there are infinitely many newforms
$f$ that are $p$-congruent to $g$, with $\lambda$-invariant equal to $n$. We
also prove quantitative results regarding the levels of such modular forms with
prescribed $\lambda$-invariant.
