The Croatian left-back, who was substituted near the end of the Europa League final defeat to Eintracht Frankfurt on Wednesday after picking up an injury in the closing stages of extra-time, will not recover in time to feature in the Hampden showdown on Saturday.
However, Giovanni van Bronckhorst confirmed that the rest of the players who featured in Seville will be available for selection.
The Rangers manager said: “We had a tough game with 120 minutes minutes in diffiult heat. We had a good recovery today.
"Borna is not available for tomorrow, all other players who featured are available for tomorrow.”
Van Bronckhorst also confirmed that Jon McLaughlin will be his starting goalkeeper for the final having featured throughout the Scottish Cup campaign this season.
“I’ve started McLaughlin in the cup campaign this year. Jon has my full confidence,” van Bronckhorst added.
"Allan [McGregor] is our number one and will fully support Jon tomorrow.”