Rodrigo's future is still doubtful, although the new coach has come up and tried to use all of his team players, but still he (Xabi Alonso) didn't even include him in his plan for the next season. So now it's time for him to leave the club because the coach didn't even give him a chance to see his potential. Xabi Alonso is trying to utilize Valverde in the right winger position, and honestly, he (Valverde) is quite good, although he is not a natural winger but is still performing as a good player.
Arsenal are the team who are like him, and they are deeply observing his situation at Real Madrid and like to sign him permanently for their team. Now it will be up to him whether he leaves the club or he still prefers to stay, although he will be trailing for the X11 sheet....
But was there any time Rodrigo future was certain. Even with the old coach Carlo Ancelotti he never had a full season of starting for Real Madrid.
There was this he isn't just good enough feeling Real Madrid had for him, they were always looking for a better replacement.
But in my opinion Rodrigo may not do well at Arsenal, I am not a hater but Arsenal loves going for players a big team considers not good enough, Gabriel Jesus is an example and he really did not do very much. But if Arsenal manages to land him I hope he does well.