The 23-year-old, who has a year remaining on his current deal, has reportedly attracted the interest of Bayern Munich, Manchester United and Real Madrid.
Dortmund want to finalise terms with the German international before the World Cup in Brazil this summer.
"Ilkay Gundogan is an important player of our squad we want to keep at our club," Watzke told Sport-BILD weekly.
"We have submitted a contract offer to Ilkay Gundogan and we are expecting to hear from him about it.
"And we are certainly not going to need to wait until the summer to see him reply."
Gundogan joined Dortmund from Nuremburg in 2011.