The Germany striker is rumoured to be on his way to Emirates Stadium but Wenger said reports that he had already had a medical had jumped the gun.
"Nothing is done. No, he has not had a medical," Wenger told a news conference. "I don't master the timing [of any announcement] because that does not just rely on us. It could happen soon, it could happen later, I don't know.
"It is always a concern [that someone else could come in for him]. The speed in which you do things is important, but at the end of the day it is always the player who makes the decision.
"It is about how much they want to join you."
Arsenal, who are on a six-match winning run which has taken them into third place in the Premier League, host Aston Villa on Saturday looking to bolster bid for a Champions League spot.
Even if they do qualify for the tournament again, Wenger ruled out big changes in the squad in the off-season.
"We will not be a lot in the market, quantity-wise I mean. Quality-wise, certainly," he said.
"We have many players out on loan and we will take a gamble if we find top, top quality of course.
"But don't forget we have many players out on loan and we have not had Jack Wilshere or Abou Diaby for the whole season.
"These players will be back and it means that number and quality-wise we are strong because we are in the position we are in without these players."
Rise of the (other) Lionesses: How one independent Women's Championship side are aiming high
‘We were in Amsterdam, so I faked a toothache to go and see Crystal Palace in the FA Cup final. I couldn’t make the replay, though – the Stones worked it out’: Rolling Stones guitarist Bill Wyman feigned injury to watch 3-3 draw with Manchester United
