The tough-tackling defensive midfielder has been linked with Chelsea and Liverpool, but has admitted he already has his eyes on a spot at the Emirates.
Xhaka currently wears the No 34 shirt at Monchengladbach and discussed keeping up that tradition with the Gunners.
"It was also here in Gladbach, with Amin Younes. And at Arsenal Coquelin has it. I've spoken with him, he would like to take another number!"
Xhaka is reportedly valued at £37million by Monchengladbach, a fee Arsenal are apparently ready to pay.
The Switzerland international has 39 caps and helped his country win the FIFA U-17 World Cup in 2009 and
EURO Under-21 Championship in 2011.
"I have a childhood dream, that is the Premier League," Xhaka said.
"Whether I play two or a hundred games in England, I would like to eventually make this childhood dream come true."
Xhaka's agent is his father Ragip, who is looking to help secure his son's dream move to the Premier League.
"Gladbach are a top club,' Xhaka told German newspaper Bild.
"But if I were someday to go to a top, top, top club, I want to play there.
"I'm not the type to be on the bench."
His first-choice pairing for the two holding roles in his midfield, Francis Coquelin and Santi Cazorla, are both currently injured.
But with question marks over the futures of Mikel Arteta, Mathieu Flamini and Tomas Rosicky, Wenger could bring in Xhaka this summer.
And while Monchengladbach would not want to sell their star man in January, Director of Sport Max Eberl has admitted he could leave at the end of the season.
"In the summer perhaps Granit Xhaka will go.
"We need a good mix between buying and selling to take the next step.
"After four successful years, we are not yet on a par with Bayern, Dortmund, Leverkusen, Schalke and Wolfsburg."
Xhaka has made 23 appearances for Monchengladbach this season, scoring three goals and receiving the same number of red cards.
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