From the few matches Xabi has played for Real Madrid, it's clear that Rodrygo isn't in Xabi's plans for next season. Therefore, Rodrygo will likely leave Real Madrid, as he's still very young and needs more playing time. Although he's been a hero for Real Madrid in key matches under Ancelotti, he's sometimes inconsistent. When he starts, he always performs poorly, but when he comes off the bench, he always makes a difference for the team.
So, it's understandable that when Xabi Alanso arrived at Real Madrid, Rodrygo wasn't high on his priority list. I personally would prefer Rodrygo to leave Real Madrid, especially as you mentioned, with so many teams still interested. Furthermore, his current asking price is still high. This would be a great opportunity for Real Madrid, as they have fresh funds to bring in other players who fit Xabi's criteria.
Club statement is that he is free to leave but the major problem will be his fees, currently the club values him above €100M which I will say at this period of his career is a very high amount for him. He fell off the picking order of Ancelloti last season and with the new manager many thought he would get the chance yet again but it is clear that Xabi Alonso is admiring players like Arder Guler and Gonzalo which means that more competition for the Brazilian.
For me he certainly must have to look for a way out because staying to prove himself might be a bit risky because the next transfer window would be in January and by then the World Cup will be coming very close, so for any player looking to make it into the world squad they simply need to stay the season on a very high level and sitting on the bench with limited playing time won’t help it