Wenger, who signed a two-year extension to his contract at the Emirates Stadium at the end of last season, has only managed to win four of the last 13 games with the Gunners.
"People tend to forget what Wenger has done for the club. I understand the frustration but everyone should stand by him, everyone should be behind [him]," Coquelin told the Evening Standard. "That's the only way results are going to turn around.
Speaking of speculation surrounding winger Alexis Sanchez, Coquelin said: "If they let Sanchez go, what I am sure of is they will have to strengthen the team, because he will be a big miss.
"For fans, they demand a competitive team. They will be upset if no-one comes in the transfer market and they let Sanchez go."