Full-back Vrsaljko, 18, is considered one of the best defensive prospects in European football, whilst 16-year-old Kovacic is a highly-rated midfielder.
Dinamo have revealed that Arsenal sent a delegation to Croatia to try and do a deal for the pair earlier this month.
But club vice-president Zdvravko Mamic has stated that he has shunned a £16.9million offer from Arsenal.
"Ten days ago we had a special visit in Zagreb," Mamic told Sportski Zurnal.
"Here were people from Arsenal and they were ready to pay 20 million Euros for Vrsaljko and Kovacic.
"But they are to stay in Dinamo, Arsenal just wanted to buy them for the future. But my answer was sharp, I said 'no' in one second."
Mamic admits he is gambling on the pair becoming even more valuable in the future.
"While I have space and no problems at our bank account I can take a risk," he continued.
"Yes, I know I'm playing with luck but so far I am winning. I believe it will be the same in the future."