The Romanian central defender played over a 100 games for Ajax before making the switch to Italy with Roma in 2003. Chivu, who can play at left-back as well, then moved to Inter in 2007 but is set to leave the club on a free transfer after his contract expires at the end of the current campaign.
Real Madrid are understood to be interested in signing the 31-year-old but De Boer hopes to persuade the player to return to Ajax instead.
"We know that Real Madrid also loves him, but we hope he will choose to return to his former house," De Boer told La Gazzetta dello Sport.
However, it appears Chivu prefers a move to La Liga with Madrid. The player's agent Joan Becali suggested as much just last month when told of possible interest from Ajax.
"Back to Ajax? I'd say he could continue playing abroad, maybe at Real Madrid," Becali said.
"I'm talking with three or four big level teams.
"He can continue playing for four or five years."
"[Real Madrid coach] Jose Mourinho knows him well. I've spoken to him, we'll see.
"The most important thing is that Cristian plays as a pure defender."
Chivu was part of Mourinho's treble winning Inter side from the 2009/10 season.