However, the Liverpool boss admitted he and his coaching staff are excited by the potential the teenager has shown in the early stages of his fledgling career.
Brewster has hit consecutive hat-tricks over the course of the last week to propel the Young Lions into the final of the U17 World Cup in India.
The 17-year-old's status as the competition's current top scorer has brought him plenty of attention from media and fans ahead of Saturday's showpiece meeting with European champions Spain.
Klopp is keen to keep a lid on any heightened expectations surrounding his talented forward, but believes he is managing a character who will take such talk in his stride.
"I don’t like to make things like this too big but it is obvious, if you score a hat-trick twice [in a row], then it's quite difficult to keep it a secret," he told reporters.
"We had Rhian around already last season but then he had a bad injury and the funny thing is I didn't see him for three months - when he came back, he looked completely different!
"He was already a kind of a man, I think I saw a bit of a beard or something like this! That will happen a lot in the next few months and years but it's really fantastic to have him around.
"He is very similar, not as a player but as a person, to Ben Woodburn. He also never thinks about which team he is playing for; he could still play U18s but now he is playing for U23s when he's here, that's a really good thing.
"He scored important goals already and is ready to make these very difficult steps in this age group. There are different ways to do that - you could score 25/30/35 goals in the U18s, or in the U23s a little bit less but he is already used to playing against adults.
"He's on a fantastic way and we are all looking forward to be part of his future.
"It's really nice with these two guys, with Ben [Woodburn] and Rhian, with Harry [Wilson] - only the offensive players - and with Dom Solanke and all that.
"We have so many more than good boys for the future but it's the future and we obviously have to sort the present and to build circumstances where these boys can come through."
- Watch: Rhian Brewster's World Cup treble against Brazil
- Rhian Brewster hat-trick sends England into U17 World Cup final
Brewster made his first appearance in the Reds' senior squad at the back end of last season, featuring on the bench during an April defeat to Crystal Palace at Anfield.
He is yet to make a first-team debut but has trained regularly at Melwood in between outings for the club's U23s side, and Klopp is patient over the prospect of handing out a senior bow.
He added: "In training, 100 per cent he was already [in the picture] - he showed up already. But it’s really not the moment to talk about [a senior debut], he doesn't expect this.
"We had a little chat after the last game [for England] and it was funny [because] I told him to save the number and he said, 'I will, for sure'. Maybe that's the first step!
"But I knew him before, he was in training very often already, but it makes not too much sense to be always in first-team training and be playing in other teams.
"You have to be part of the team where you are playing as well. It's all good, the future's bright, we are responsible, he is responsible and unfortunately you as well, that’s the deal.
"I think we are all are interested that these skilled boys can come through at the end, can become Premier League players, can become players for the Three Lions and stuff like this - that’s what we all want. For that, we have to create the situation."