The 26-year-old is understood to be keen to join the Gunners in the summer, with suggestions an £11million deal had already been agreed with Cologne.
The Poland-born forward would certainly add some experience to the Arsenal attack in support of 32-goal captain Robin van Persie, having played at the World Cup twice and also in two European Championships.
Wenger, though, maintained at Friday's press conference ahead of Monday night's Barclays Premier League clash against Newcastle, that the transfer was not yet a done deal.
"I have not a lot to say about that because if we sign a player we will inform you. At the moment we have not signed anybody. There is still no agreement.
"The distance to the agreement is not always linked with the signing. When you are very close you are quickly very far again, but Podolski is a good player, I wouldn't deny that."
When pressed on the possibility of completing a deal by visiting German media, Wenger suggested there could be some movement "in the next two or three weeks", adding: "anything is possible".
Van Persie has yet to agree a contract extension, which runs until the summer of 2013.
Manchester City manager Roberto Mancini has fuelled speculation of another big-money raid on the Emirates Stadium for the Holland striker, whom he hailed as "one of the best strikers in Europe".
Wenger insists such comments must be taken with a pinch of salt.
"If you are to have one quality it is not to be unsettled because what is said of you by other clubs is part of the job," the Arsenal boss said.
"What is important is how much you want to win the next game and all the rest is basically not important."
Wenger added: "Robin is the leader of our club and I would just like to specify that he is not at the end of his contract.
"He is under contract until 2013. So you do your job, focus on your job which he does very well and enjoy your games."
Brazil full-back Andre Santos could be in the squad on Monday, having recovered from the broken ankle which sidelined him since December.
Wenger is also hopeful of a positive prognosis for England midfielder Jack Wilshere, who has not played at all this season because of his ankle problem.
"The progress in training is very good but what you do not want is to have any setbacks," Wenger said.
"The setback could be a bad scan, but the clinical signs are all very good when you look at his ankle, it is not inflamed and it is normal progress now.
"But we are a bit over cautious with him now because of the setbacks we had before."
Wenger maintained he was "surprised" to have been charged with improper conduct by UEFA after confronting referee Damir Skomina following Arsenal's 3-0 win over AC Milan on Tuesday night.
If the case is proved and Wenger is given a touchline ban, it will be the third time within the space of a year he has been given such a sanction by Europe's governing body.
With their Champions League hopes now over following the 4-3 aggregate defeat, Arsenal will be out to continue their momentum on Monday night, building from successive wins over third-placed Tottenham and at Liverpool.
"We want to finish as high as we can," Wenger said.
"It will be difficult because we know that we face a massive battle there with Tottenham, Chelsea, Newcastle and maybe Liverpool.
"We have created the momentum but we know until the end of the season that every game will be a massive one for us.
"What is at stake is to be in the top four, if possible in the top three, but it will be difficult for everybody."