I was originally going to ask whether this statement has consistency strength over $\mathsf{ZFC}$: there exist transitive sets $M,N$ with $N\subseteq M$ and a nontrivial elementary embedding $j:M\rightarrow N$ (moves some ordinal). Then I realized Ehrenfeucht-Fraisse game can probably be used to show the existence of nontrivial $j:\kappa\rightarrow\kappa$ for any cardinal $\kappa$. So my question is what if we require $M$ to satisfy some basic set theory, like pairing+union? What's the "strongest" theory $T$ for which such a pair satisfying $T$ is relatively consistent with (or provably exists in) $\mathsf{ZFC}$?
As a first attempt, if $j:\kappa\rightarrow\kappa$ is an elementary embedding, and $M$ is the closure of $\kappa$ under pairing and union, does $j$ extend to an elementary $j^+:M\rightarrow M$?
Remark: Since ordinals are definable in any transitive set (an ordinal is a transitive set of transitive sets), $j$ must restrict to an elementary embedding of $M\cap\mathrm{Ord}$ into $N\cap\mathrm{Ord}$, so $M\cap\mathrm{Ord}\subseteq N\cap\mathrm{Ord}$. Since I also assume $N\subseteq M$, we have $M\cap\mathrm{Ord}=N\cap\mathrm{Ord}$. Without requiring $N\subseteq M$ there are easy examples in $\mathsf{ZFC}$. It seems the exact consistency strength of $j:M\rightarrow M$ where $M\models\mathsf{ZFC}$ is known, but my question is more about how much set theory can we make $M$ satisfy if working in $\mathsf{ZFC}$.