World number one Alcaraz eased into the third round in New York on Thursday with a 6-3, 6-1, 7-6 (7/4) win over South Africa's Lloyd Harris.
He is aiming to become the first man to defend the US Open crown since 2008, when Roger Federer hoisted the trophy for a fifth consecutive year.
None of that has stopped him keeping an eye from afar on the early-season fortunes of Madrid, who lead La Liga after three games - largely thanks to the goalscoring exploits of Bellingham.
"I'm really happy to watch him play in Real Madrid," said Alcaraz of the England star signed from Borussia Dortmund for over 100 million euros (£86.5 million) this summer.
Bellingham has scored four times in three games for the league leaders, drawing the admiration of his fellow 20-year-old, Alcaraz.
"He's such a great, talented player, one of the best in the world," said the Spaniard.
"I'm sure that he's going to be the best player in the world in that position. I'm just really, really happy to have him in the team.
"I talk a little bit with him. He's such a great person, as well. It's a complete guy."
Alcaraz had one minor blip in the third set against Harris but has been largely untroubled through his first two matches at the US Open after his opening-round tie was cut short by injury.
"I think I played a great match," said Alcaraz. "If I have to keep something from the match I'm going to keep the level from the second set."
"I had to forget that break (in the third set). I did such a bad game. I had to forget it and focus. To be able to win in straight sets is really important to me in the first rounds."
The 20-year-old, who will lose the number one ranking to Novak Djokovic after the tournament, plays Britain's 26th seed Dan Evans in the last 32.
"It's going to be really tough. I will play my best. I will have to return very, very well if I want to win that match. Let's see how it's going to be," said Alcaraz.
Asked about potentially emulating Federer 15 years on, he replied: "It's a motivation. It would be amazing to be able to share that achievement."
