The 23-year-old Japan international has attracted the interest of several top clubs during another memorable season in which he has scored 12 goals and set up another seven to help Dortmund to the top of the Bundesliga.
Snapped up for just 350,000 euros from Japan's second division in 2010, Kagawa is now valued at around 15 million and Dortmund are eager to extend his contract which runs out next season.
Premier league Chelsea and Manchester United are reportedly among the clubs interested in the skilled player.
"We have had talks with Shinji. From our side we have made him an offer and now Shinji has time to think about it and decide," Watzke told Reuters in an interview.
"Then we could have a talk after 14 days or so. That is if it is necessary. We have not given Shinji any ultimatum."
"He could say yes now but if he wants to talk about it then we can talk. Then a decision can be taken in the coming weeks. But I am confident that Shinji will stay with us," said Watzke.
Watzke is credited with steering Dortmund away from financial collapse some seven years ago and guiding the club and 1997 Champions League winners into the most exciting sports brand in Germany in the past two seasons.
But even Watzke is surprised by his team's stunning season where they have gone 22 straight league games without a loss and are top with a three-point lead over Bayern Munich and six games left to play.
"Yes, absolutely," he said when asked whether the team's second consecutive dominant season was unexpected.
"The team showed the same consistency as last year only this season we had a rival. Bayern were not that strong last season. But this season both teams are playing on a very consistent level and are likely to end up with more than 70 points each," he said.
"That is very unusual and especially since at the start of the season we expected Bayern to run away with the title," said Watzke, who extended his own contract to 2014.
Dortmund, who have all but secured qualification for the Champions League group stage next season, will also face Bayern in the German Cup final in May.
"After 28 games we can say we are completely satisfied with the current season. It looks very good, we look set to go in the Champions League group stage," said Watzke.
"If someone had told us last year that we would be in the top three or four and that we would be in the German Cup final as well we would have thrown a party before the season."
"But if you are within touching distance of two titles then you want to win at least one," he said.
Watzke said planning for next season had yet to start given possible changes to the squad that could also include the departure of Paraguay international Lucas Barrios, who lost his starting spot to Dortmund top scorer Robert Lewandowski.
"We have not dealt with next season for one second yet. We will start doing that once we know with which players we will go into the next season," he said.
"Lucas, concerned about losing his starting spot, was told back in the winter that if he wanted to leave then Dortmund would put no obstacle in his way. But whatever will happen will happen after the end of the season," Watzke said.
Dortmund's run towards their second straight title will face its biggest challenge in the coming weeks with back-to-back matches against Bayern, bitter Ruhr valley rivals Schalke and fourth-placed Borussia Moenchengladbach.
The match against Bayern on April 11, likely to be a title decider, could also mark the return of talented midfielder Mario Goetze, out since January with a groin injury.
The 19-year-old Germany international has returned to training but it is still unclear when he will be fit to play.
"That is up to the coach to decide but we would be wise not to put him under too much pressure," said Watzke.
As for the future of the booming Bundesliga, which is on the brink of a bigger broadcasting rights deal and has the highest average per game attendance in the world with over 42,000 spectators, Watzke said the outlook was bright.
"The 4-4 draw between Dortmund and Stuttgart last week was proof that the Bundesliga does not need to worry about its future," he said of the exciting encounter in which both teams scored in the final minutes.
"I am constantly asked about this game. It was the most magnificent advertisement for the Bundesliga and I am certain the league will have an extremely positive development over the coming seasons," he said.