The Ivorian winger is battling to get back in his manager's favour after a series of inconsistent displays and a red card against Leeds for a clash of heads with Ezgjan Alioski.
Arteta is said to have taken Pepe to task in the dressing room after Sunday's 0-0 draw, as his dismissal left the Gunners playing the majority of the second-half with ten men.
He also called the forward out publicly for his 'unacceptable' conduct, with the 25-year-old apologising for his sending off on Monday.
"I let my team down at a crucial time of the game and there are no excuses for my behaviour," he said.
"I am deeply sorry and would like to apologise to the fans, my team-mates, my coach and everyone else at the club."
According to The Times, the incident has added to Arteta's doubts about Arsenal's club-record signing, and whether he has what it takes to succeed at the Premier League side.
Pepe has started only two league matches this season, scoring once, and patience is wearing thin over whether the winger signed from Lille in August last year will live up to his huge transfer fee.
The Times claim the Gunners will NOT look to loan out Pepe in January in a bid to regain his confidence and address his disciplinary issues, however, with Arteta set for talks with the forward on Wednesday.
The Arsenal boss prefers to hold private meetings with his players to address disciplinary problems, which was the case when Matteo Guendouzi was exiled for his poor attitude in July, and also when summer signing Willian travelled to Dubai without permission during the recent international break.
But, unlike Guendouzi who has been sent to spend the season on loan at Hertha Berlin in the Bundesliga, Pepe is set to stay put, largely because of the Gunners' busy schedule, both domestically and in the Europa League.
Instead, they must hope he recaptures the form that led to him scoring 22 goals and claiming 11 assists in his final season in France.
Pepe will be part of the Arsenal squad that travels to Norway on Wednesday - a day earlier than usual - for their Europa League group stage clash against Molde on Thursday.
His dismissal against Leeds was Arsenal's fifth red card since Arteta took charge 11 months ago, with ill-discipline threatening to derail the club's progress under the Spaniard.
Pierre-Emerick Aubameyang saw red against Crystal Palace in January, David Luiz was dismissed against Chelsea and Manchester City, and Eddie Nketiah was sent off for a high tackle on James Justin, of Leicester City, in July.
Arsenal are 'unlikely' to loan out Nicolas Pepe in January, but boss Mikel Arteta has planned talks with the £72million flop over his poor form and 'unacceptable' conduct.