The 21-year-old joined Inter in 2013 and was described by Mancini last season as Inter's future.
Commenting on Kovacic's widely reported move to Real, Mancini told Inter's website: "There are rules that have to be respected. I don't think anyone wanted this to happen but we have the Financial Fair Play regulations to follow.
"We're all sorry about it: myself, the president, the management and the players themselves," he added after Sunday's 0-0 draw with AEK Athens in a pre-season friendly.
Mancini, whose side finished a dismal eighth in Serie A last season and will miss out on European football this year, has already seen Swiss winger Xherdan Shaqiri leave for Stoke City.
"(Kovacic) and Shaqiri both left for the same reason, but now we'll try to bring in the sort of players we need and hopefully we'll be able to do it quickly because the league starts next week," he said.
In May, Inter agreed to pay a fine of up to 20 million euros in a negotiated settlement for breaking UEFA's break even rules known as Financial Fair Play.
UEFA said Inter would automatically pay six million euros, which would be deducted from their European competition revenue, while payment of the remaining 14 million was conditional.
(Writing by Brian Homewood in Berne; Editing by Peter Rutherford)
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