The Mirror says Rovers are thrashing out terms with Hajduk Split and the 21-year-old has spent the past two days in Lancashire.
Rovers put in an offer last night and Split are mulling it over. But there is also a rival bid from Porto, who want to buy a "half share" in Kalinic.
Blackburn appear to be in pole position and Allardyce hopes to have Kalinic in his side for the Premier League kick-off with Manchester City.
But because he has just three full Croatia caps a work permit application would go to an appeal, although Rovers are confident it will be granted.