According to the Daily Mirror, Allardyce has started talks with the hitman and his club Hajduk Split and hopes to complete the deal quickly, following the players decision to walk out on a deal with Premier League rivals Portsmouth.
Kalinic, 21, is due to return to Croatia today to play for Hajduk at the weekend, but Rovers hope they can get him to sign a four-year deal.
Hajduk are willing to take the cash, but they have been left fuming that Portsmouth were not willing to back up their interest.