"Shanghai has tabled a very substantial offer to Nathan Burns and the indications are he is happy to accept it," Alilovic told the website.
"But the situation is that Shanghai are not willing to match Wellington's asking price, which for a player soon to be out of contract they believe to be somewhat unrealistic. The figure will need to be in a far lower price than that."
Burns, who is playing for Australia in the ongoing Asian Cup, has been a revelation for the Phoenix this season after arriving in Wellington in need of a change.
The 26-year-old has scored a league-leading 10 goals in 12 games as part of Ernie Merrick's high-octane attack where Burns has linked well with fellow new recruit Michael McGlinchey.
Burns told the Herald last month he owed a lot to Merrick who helped spark his career revival that lead to his recall to the Socceroos.
"He's a good coach and we are playing a good style of football, which helps a lot," Burns said. "It gives you that freedom to play an attacking brand of football that I want to play so it definitely helps with the way Ernie's coaching at the moment. And he's still pushing me and doesn't give me any days off."
The Phoenix, who are in second place on the A-League ladder, will return to action on February 2 when they meet the Brisbane Roar across the Tasman.