The Catalan club has endured a mixed start to the season under the former Netherlands boss, who took charge after Quique Setien was sacked in the wake of the 8-2 thrashing by Bayern Munich in August.
Barca started its 2020-2021 Champions League campaign with a 5-1 win over Ferencvaros, but it has taken only seven points from five matches in LaLiga.
Barca lost the first Clasico of the season 3-1 at home on Sunday (AEDT) to Real Madrid, which was reeling from consecutive defeats to Cadiz and Shakhtar Donetsk, with Koeman's men failing to muster a single shot on target after a Lionel Messi effort in the 24th minute.
Koeman, however, has urged fans to be patient as he attempts to turn Barca back into a side capable of winning top honours.
"When I was speaking with Barca to be coach, I knew it wouldn't be an easy challenge because of the changes we want to make, because of the COVID-19 issue, the economic issue, but the challenge is very big," he told reporters on Tuesday.
"I just ask for calm, for time, because we're changing a lot of things. In the game with Madrid, in 60 minutes, the team was very good."
Koeman is preparing his side to face Juventus on Wednesday in what will be opposite number Andrea Pirlo's biggest game in his short coaching career.
Pirlo, who took on his first head coach job prior to the start of the season, may be without Cristiano Ronaldo, who has been in quarantine after testing positive for coronavirus while with Portugal.
Koeman hopes to see Ronaldo available despite the formidable task of facing the Champions League's all-time leading goalscorer.
"It's a game between two great teams who want to go very far in this competition. It's important that we get a good result, with the football we want to play," he said.
"It's an important game for everyone and beautiful to watch. We always want the best on the pitch. Ronaldo is a great player. I don't know if he'll be able [to play]; it's not my problem, nor my decision. But hopefully, he can be there.
"It's easier to give an opinion on Pirlo as a player. He was a great player in midfield and he gave a lot of success to various teams in Italy. He's had little time as a coach and if he's already the coach of a team like Juventus, it's for a reason."
Koeman could reportedly face disciplinary action in Spain for his comments after the Clasico, in which he suggested VAR has been used to penalise his team this season.
"I don't regret my comments after the Clasico," he added. "I said I wouldn't speak further on the subject and the most important thing is the game tomorrow. It's important to play well and get a good result."