Can I play Pouncing Rengar in base as a reaction on my opponent's turn?

Yes, you can play Rengar, Pouncing to your base as a reaction on your opponent’s turn.

According to Rule 813.1.c.1 and the [Reaction] keyword on the card, Rengar can be played during Closed States on any player's turn. Additionally, Rule 813.3.a specifies that units with [Reaction] still follow standard playing restrictions unless overridden; standard rules permit you to play units to your own base at any time you have the timing and resources to do so.

Since [Reaction] grants you the permission to play him during a Closed State (such as when your opponent has a chain on the board), you can play him to your base at that time.