The Liga side made an offer of €30 million (£27m/$35m) for the 21-year-old and were willing to let him stay at Ajax on loan for another year.
Ajax held firm however, and were not willing to let their prized prospect leave - even for a higher offer.
The Spanish side were hoping to avoid the inevitable bidding war for De Jong.
Premier League clubs Tottenham and Manchester City are also rumoured to be hot on the trail of the Dutch star, and Barcelona feel they are losing some of their financial might and could be left out in the cold if a bidding war over De Jong actually occurs.
De Jong himself hinted at wanting a move to Barcelona, telling Ziggo Sport that he would like to play alongside Lionel Messi in the future.
“If you end up in a club of a higher level, you will play with world-class players anyway," he said.
“But I would really like to play with Messi someday.
“We were on vacation in Spain and my grandfather bought me Messi's shirt with the 30, his shirt number then.
“My grandfather knew that I liked Messi and I wore that shirt all the time.
“Of course I would like to play with him. He's already 31, so I have to hurry up.”
Ajax are off to a flying start this season, having won four of their opening five matches and drawn the other.