The stalemate left the Gunners five points adrift of Barclays Premier League leaders Leicester, and means they have won just once in the competition this year.
Hosts Arsenal had 11 shots on target but have now dropped nine points from their last four games.
Several newspapers reported Wenger was involved in a spat with both referee Lee Mason and Southampton counterpart Koeman in the tunnel after the final whistle.
According to the Daily Mail, Wenger approached Mason and said: “It’s always the same with you.”
Koeman, who had overheard the Frenchman, then reportedly said to Wenger: “It’s always the same with you.”
Southampton’s Dutch boss also claimed Wenger was blaming the referee for his own team’s failings, according to the Daily Mail, saying: “You had 10 chances to score and didn’t take any of them. Instead you take it out on them (the officials).”
Saints goalkeeper Fraser Forster put in a superb display to frustrate the Arsenal boss.
Forster kept a fourth consecutive clean sheet following his return from a serious knee injury, with the England international saving well from Mesut Ozil, Alexis Sanchez and Olivier Giroud among others.
The Frenchman conceded later on that his side need to find their form of late 2015 to get back in the title race, with only a 4-0 reverse at Southampton on Boxing Day blotting their copybook before Christmas.
“We’ll see at the end of the season. It’s too early to say that,” he said when asked if he was worried that Arsenal were starting to tail off.
“It’s true that we’ve had a stuttering start in 2016. Our January was bad, starting from the last minute at Liverpool, and then at Stoke. Where we especially have a big deficit is in our two home games.
“We have to catch that back and we have a difficult programme, especially away from home. We need to respond quickly away from home now.”
Having added only midfielder Mohamed Elneny to his ranks in the January transfer window, Wenger was again forced to defend his decision not to bring in any firepower.
“If you knew a world-class striker who could have strengthened our team, you should have told me before the transfer deadline was over,” he replied when asked if he should have signed a forward.
“They are all at big clubs and are under contract. There were none available. We can score goals, we have shown that before.”
Koeman hailed shot-stopper Forster as a “magic man”.
He said: “Arsenal are so good defensively with a lot of creativity and that’s difficult then, of course, you need a keeper that’s magic and he is magic tonight.
“He is with the national team and he is one of the best goalkeepers in the Premier League and maybe in Europe.”