According to the Daily Express, Arsenal are prepared to spend up to £20m to land the Manchester United target, who helped Dortmund reach the Champions League final last season.
The 22-year-old has been identified by both United boss David Moyes and Arsenal manager Arsene Wenger as an ideal Premier League midfielder.
And Wenger believes he could be the perfect long-term successor to Mikel Arteta in the heart of the Gunners midfield.
Despite Arsenal's willingness to test the water with an offer, the report claims Gundogan's father has cast major doubt on any potential deal, insisting that his son wants to stay in the Bundesliga.
Speaking to German publication Kicker, he said: "I am keen to ensure he [Ilkay] remains with Borussia Dortmund.
"He is doing fine so it's a bit early to move to a club in England or Spain at this point in time."
And he even suggested the Manchester United and Arsenal transfer target could put pen to paper on a new long-term Dortmund contract.
"Borussia Dortmund will be coming up with an offer of a new contract soon," he continued.
"We will be able to oversee things much better than."
The midfielder's current deal expires in the summer of 2015, and Dortmund are keen to prevent a repeat of the Robert Lewandowski situation.
Another one-time Manchester United and Arsenal target, Lewandowski looks set to run down the remaining year on his Dortmund deal before moving to Bayern Munich.