Chile international Alexis reacted angrily to being substituted after completing the scoring in the Gunners' 4-0 win at Swansea City on Saturday, but Wenger has now suggested that his Stamford Bridge counterpart Antonio Conte has far greater concerns with his own star striker.
Costa sat out Chelsea's 3-0 triumph at Leicester City on the same day, when the leaders extended their advantage at the summit to seven points.
The official reason for his absence was a back complaint, but the Spain international's exclusion came amid reports of a training ground row with Conte, framed by speculation that Costa is the subject of a mega-money offer from Chinese Super League side Tianjin Quanjian.
"I'd rather have someone who always wants to be involved than someone who doesn't want to play," Wenger told reporters.
"I have enough problems —I don't need a Costa problem on top of that.
"Alexis is a guy who wants to fight everywhere, sometimes where he shouldn't. But that is a part of his game. All my players want to play for every minute but for me this is not a story."
Spurs manager Mauricio Pochettino also gave his take on the Costa situation and was happy to report no similar rumblings of discontent at White Hart Lane.
"Have there been any offers from China for my players? No," he told reporters.
"My reaction to the Costa story was indifference because, luckily, it is not my problem.
"What can you do when an offer like this arrives from a country that is like this?
"It's crazy money. It looks like they broke the market. If the offer to Costa is true, it's unbelievable, unbelievable. But we will see what happens."
Pochettino added: "We are together, every player is happy and part of our Tottenham family, and we have a very good atmosphere around the club."