The Austrian international has also been targeted by Everton and Middlesbrough.
Grazer AK are considering Fulham's bid of around £2m and will let the Cottagers know within the next 2 weeks whether they are prepared to do business.
GAK general manager Hannes Weninger said: “We have been asked to make a final decision within the next 14 days.”
“We don’t want to lose him, but Rene has already told us that he would love to accept an offer giving him the chance to play in the Premier League."
GAK general manager Hannes Weninger said: “We have been asked to make a final decision within the next 14 days.”
“We don’t want to lose him, but Rene has already told us that he would love to accept an offer giving him the chance to play in the Premier League."
Meanwhile, Coleman is about to sign Australian international Ahmad Elrich.
Elrich has not played club football since walking out in November on Korean club, Busan Icons.
However, it is thought that Coleman may have to pay around £300,000 to sign the former Parramatta Power star.
Elrich has already completed his first training session with the west London club after being referred to the Craven Cottage management by Liverpool legend Ian Rush.
Should Coleman secure the 23-year-old Socceroos winger, he will beat long-time admirers Blackburn to his signature.