The 21-year-old could cost Blackburn up to ¦pound;12million, according to Hajduk Split.
Kalinic was also targeted by Portsmouth this summer but Rovers had funds in place following the sale of Santa Cruz and the youngster is set to spearhead their attack in the new campaign.
It was the second piece of transfer business for Rovers on Monday afer boss Sam Allardyce signed striker Franco Di Santo on loan from Chelsea for the first half of the season.
Allardyce has been determined to strengthen his attack following the departure of Roque Santa Cruz to Manchester City and Di Santo will give him a fresh option.
The 20-year-old Argentinian made only a minimal impact on the first team at Stamford Bridge after signing from Chilean side Audax Italiano in January 2008.
He will stay with Blackburn until January under the terms of the deal completed on Monday night.
Rovers will then have first option on extending the deal providing Chelsea are willing to let him stay on loan for the remaining months of the season.
Di Santo could make his first appearance for Blackburn in Wednesday's home friendly against Hibernian.