According to media reports, the attacking midfielder will cost a fee of €11 million.
Inter also announced the acquisition of Argentine-born winger Ezequiel Schelotto from Atalanta and Swiss-born Serbia midfielder Zdravko Kuzmanovic from Stuttgart.
Kovacic was born and raised in Austria but joined Dinamo as a teenager and became the youngest scorer in Croatian league history when he scored his first professional goals against Hrvatski Dragovoljac at the age of 16 in November 2010.
He won two league titles with Dinamo and has appeared for Croatia at various age levels including Under-21.
"It all happened really quickly," Kovacic told the club website. "I was training with Dinamo and my coach told me I was coming to play for Inter.
"It was like a dream for me and I can't really describe what I feel at the moment. They're one of the biggest clubs in Europe...I hope to stay here for a long time."
Former Fiorentina midfielder Kuzmanovic is returning to Serie A after three-and-a-half seasons with Stuttgart.
"I'm delighted to have joined a great club like Inter, that the deal went through and also that I'm back in Italy," said Kuzmanovic who appeared for the Swiss Under-21s but then chose to play for Serbia at senior level and has won 46 caps.
Schelotto, who has Italian citizenship and made his debut for his adopted country last year, has been at Atalanta since 2010.
"I'm glad I've joined Inter, the biggest club in the world," he said.
Inter, who won the Serie A, Italian Cup and Champions League treble under Jose Mourinho in 2009-10, are fourth in Serie A with 40 points, nine behind leaders Juventus.