In his programme notes before Sunday's Premier League clash at Old Trafford, United boss Jose Mourinho said that Everton "want to at least secure a top-four position this season" having spent £140m in the summer, a comment which seemed to anger Koeman.
Speaking after Everton's 4-0 loss to United, Koeman reacted to Mourinho's statement by saying: "Be realistic!
"I read the programme of United and my colleague and I talked about 'Everton spending £140m and they need to go for the top four'.
"Sorry. If anybody in this room or outside sees something realistic [there], in what is possible for Everton... please, come on. Be realistic.
"Let's talk at the end of the season. I am not happy how we have started the season, but let's be realistic about Everton. Let's start with everybody. Fans, press... we need time. But it is difficult in football."
Asked about his future at the Merseyside club after a string of bad results, the Dutchman said his side need to adapt to the Premier League after a major overhaul during the summer.
"Every manager in life has doubts, there is nobody who doesn't have doubts as a manager," he added. "And if you don't win, you know you have a big number of players in your squad, and you have doubts about what system to play.
"This is normal. Of course, I ask myself the question 'why?' Why? But OK. If [Henrikh] Mkhitaryan got one year to adapt to the Premier League, we have eight new players... maybe I also give them one season to adapt to the Premier League."
Three of United's four goals on Sunday came in the final 10 minutes and the scoreline, arguably flattered the home side, with Koeman adding that he "wasn't so impressed" by Mourinho's side.
"Sometimes in life and in football, you get more than you deserve," he said. "I think today we got less than we deserved. But it is all about scoring and not making mistakes, like the second goal.
"I wasn't so impressed by United. Perhaps that's strange after a 4-0 defeat, and yes, they were really clinical - but we didn't deserve this bad defeat. But hey, that is football."