But he urged them to keep expectations “in the right context” as they face a tough part of the season.
The club’s chief executive and deputy chairman sent a message to supporters ahead of the 1-0 defeat by Crystal Palace.
The match saw boos from some home fans which appeared to be directed at head coach Fabian Hurzeler.
There were also calls for him to be sacked.
Barber’s words in the match programme revealed he was aware of discontent even before the Palace game.
He wrote: “I have also read a range of comments and criticisms of late, some constructive, some less so. One thing I would point out is we are, of course, aware of fan sentiments.
“As football fans ourselves we will also at times feel similar frustrations and emotions. It is, after all, a game of emotions.
“As such, I won't - and we won't - shy away from that criticism or ignore it, but we won't always agree with it either.
“One thing I promised fans when I first came here as chief executive, was transparency and dialogue, and we will continue to provide that - attending fans’ regular forums, answering questions, addressing concerns, replying promptly to emails and letters and, yes, where appropriate or necessary, challenging unfair or inaccurate narratives.
“Equally, I would hope you are more than aware by now that we will never take our fans’ commitment for granted, and all of us at the club appreciate the efforts our fans make to support the team.
“We know the fans want us to see us challenging for European places and that remains our ultimate aim.
“But we must also frame those ambitions in the right context and recognise there are 19 other teams in the Premier League, many with similar aspirations.
“There are no easy matches at this level of football and certainly no guarantees.
“There is still a lot of football to play and points to win and to increase our chances of reaching our aims, we do need the support of all fans at all times.
“The players, coaches, and staff. Really appreciate your encouragement. It matters and it means a lot.”