The 15 year-old midfielder has been training with Rice and Arsenal's senior squad this season.
Discussing the breakthrough of Ethan Nwaneri, 18, Rice also raised the prospect of Dowman's eventual senior introduction.
The England international told the BBC: “It is unbelievable times. I can’t talk highly enough of Ethan as well. Just wait until he hopefully gets into the team, the lads get to see him, because he is a special talent.
“Max is the best 15-year-old in the country. I have had so many texts about Max recently where people have watched him and said: ‘Wow, what a talent’. So he is one to watch out for. But I think at 15 there is still such a long way to go. You can be the best in the country at 15, but it doesn’t necessarily mean at 18 you are going to be.
“So you need to be hungry, keep working, keep pushing. I speak with Max a lot and he knows that. And he has got such a good family around him. He is unreal.
“Then obviously with Myles (Lewis-Skelly) as well, Arsenal are blessed to have three players that can potentially have long Arsenal careers. Luckily enough, I have a really good relationship with all three of them because Max gets to train with us quite a lot.”
Rice also revealed: “Those three I have kind of taken under my wing a little bit because I have been in that position before and I had older players help me out.
“I get on really well with all of them and they are all really, really comfortable around me, which is all part of having a good relationship.”