The Lance Armstrong case in cycling and this week's revelations of widespread doping in Australian sport have again put drugs in sport high on the public agenda, and Wenger does not think football can consider itself immune from such underhand practices.
"Honestly, I don't think we do enough, because it is very difficult for me to believe that you have 740 players in the World Cup and you come out with zero problems," said Wenger.
"Mathematically that happens every time, but statistically, even for social drugs, it looks like we could do better and go deeper."
Wenger continued: "When you have a doping control at UEFA, they do not take blood, they take only urine. I have asked many times (at UEFA meetings) in Geneva (for that to be changed).
"Sometimes you have to wait for two hours after the game, so blood could be a lot quicker. We could go much deeper into control. I hope we (in England) do not have a big problem with doping, but we have to try to find out and see how deep we can go into control.
"I would support it. UEFA is ready to do it, but it poses some ethical problems because everyone has to accept that they will check the blood and not everybody is ready to do that."
Wenger feels it is time for those who are found to be involved in doping to give up all they know. Dr Eufemiano Fuentes is currently on trial in Spain following his arrest as part of the Operation Puerto doping case, with the former president of Spanish side Real Sociedad, Inaki Badiola, admitting earlier this week that during 2008 the club made payments to Dr Fuentes for the supply of medicine to the team. Dr Fuentes denies any knowledge of a conspiracy to dope.
"What I am concerned about in the trial of the Spanish doctor is that he is in front of the justice (system) only to see how he did doping - they are not interested at all in whom he has doped," Wenger said. "They have found packets of blood, but they don't even ask to whom does that belong?
"The only thing we know is that the former (president) of Real Sociedad, who came out, is involved in that. The justice should go deeper. When you look at the functions of this doctor it is quite scary - he was involved in the Olympic team, football team, cycling team."