Unai Emery has been offered public backing by the Arsenal hierarchy, but warned results must improve.
Emery's position is under scrutiny after a run of just two wins from their last 10 Premier League games.
However, Arsenal head of football Raul Sanllehi and managing director Vinai Venkatesham say they have no immediate plans to make a change of manager.
"We firmly believe Unai is the right man for the job," they told Arsenal staff at a meeting.
"We are as disappointed as everyone else with both our results and performances at this stage of the season.
"We share the frustration with our fans, Unai, players and all our staff as they are not at the level we want or expect. Things need to improve to meet our objectives for the season.
"We are all working intensively behind the scenes to turn things around and are confident we will.
"We never take our fantastic support for granted. We hope we can all stick together and get behind the team in this challenging period, as together we are stronger."
Xhaka 'hurt' by hostile criticism
Meanwhile, Arsenal midfielder Granit Xhaka says he has been hurt by the "extreme hostility" directed towards him from the supporters, but has promised to prove his worth.
The Switzerland midfielder has not played for the club since.
"It was very hurtful and frustrating," Xhaka told Swiss newspaper Blick.
"I can't understand a reaction like that even now, especially the vehemence of it and the extreme hostility directed against me."
Xhaka was booed as he walked off the pitch as he was substituted against Palace, prompting him to cup his ear, take his shirt off and head straight down the tunnel to the dressing room.
"When my shirt number lit up on the fourth official's panel and our own fans broke into gleeful jubilation, that hit me very hard and really upset me," he added.
"Insulting and swearing at your own captain will cause upset and a bad atmosphere for the team you are actually supposed to be supporting; that makes no sense to me and weakens the team's spirit."
Arsenal manager Emery said last week that he was unsure whether the midfielder would play for the club again, adding that "he was not ready" to return for Saturday's match at Leicester, which the Gunners lost 2-0.
Xhaka, though, says he is fully committed to the club and is ready to move on from the incident.
"I've been 100% behind the club and my role as a player since I came here," he said.
"I'm proud to be playing for this big club. I'll continue to stay positive, give my all to an even greater extent and prove that I'm an important part of this great team.
"Last week in particular was a very special, emotional experience for me but I'm doing very well again, I've trained well this week and am looking forward to my next assignments."