Before Chelsea plans to sign Bellingham, Real Madrid already tried to sign him first. So, Real Madrid isn't joining the race but Chelsea is joining race.
I agree that Bellingham should prefer to move to Real Madrid. Chelsea has no power to sign Bellingham, Real Madrid is the favorite team. In my opinion, Chelsea seems impossible to sign Bellingham, they will have no chance. I suggest Chelsea to look for another player.
Chelsea can still sign him if they are so serious about it. Obviously, they would have to offer him a lot of wages and also make sure they qualify for next season's Champion's league tournament. Real Madrid has a very packed midfield and i feel a play like Bellingham may be hesitant to go there because of the likelihood of ending up warming the bench.