The 20-year-old is highly-rated at Anfield but his chances have been limited so far – just four cup appearances – and with Diogo Jota arriving in a £43million deal last month his opportunities were further reduced.
After an impressive loan spell at Swansea in the second half of last season, where he scored 11 goals in 22 appearances, Klopp felt there was a need for the striker – who missed more than a year with a serious knee injury – to play full-time.
That is why they have agreed a £23million deal with Sheffield United, although to protect themselves have inserted a buy-back option active for three years plus a 15 per cent sell-on fee.
“He couldn’t make it here so far but he has made big steps in his development,” said Klopp.
“Rhian was really seriously injured and first we had to make sure he was ready again for all the demands of professional football and he is, 100 per cent.
“He played a really good half-season at Swansea, he was really impressive, he came (back) here and scored for us in pre-season and you could see he made big steps.
“If Rhian would be now 25, 26 I would say ‘OK, you can stay and work for your chance and then don’t say I didn’t say you couldn’t stay’.
“But it is a different situation when you are younger, (it is important) a player like Rhian in this position up front makes now the next steps and uses the time he lost during the serious injury – so it is possible a move to another club would make sense.”
While Brewster is on the way out Klopp is ready to welcome back captain Jordan Henderson, who has been missing with a knee injury since coming off at half-time of the victory at Chelsea on September 20 with a knee injury.
The midfielder returned to individual training this week and although he will participate in some group sessions Sunday’s match at Aston Villa may come too early for him.
However, having been picked by England manager Gareth Southgate Klopp is in the unusual position for once of wanting one of his players to get more minutes with the national team.
“I was in close contact with Gareth Southgate and it is clear that he will use the time with national team for getting match-fit,” added the Reds boss.
“Hendo had four or five days normal pre-season before he played his first game for us because of the situation we were in.
“He played against Chelsea and got this little injury, which was not serious, but he has been working on his physical things already for a few days but now (it will be) more football specific with us and then the national team.
“He will have minutes for England, which is then a proper win-win situation which I’m really happy about.”