Belgium international Vertonghen has been linked with both Manchester City and Arsenal, and recently hinted he would consider a move away from Amsterdam.
He has a contract with Ajax until 2013, but De Boer is optimistic the defender can be persuaded to extend that.
"I asked him to sign and he was positive about it," De Boer told De Telegraaf.
"Danny Blind (Ajax's technical manager) will soon hold talks with his agent.
"Jan has indicated he would move to any club, but for now he is at Ajax and seems to be happy here.
"I've given him a picture of what can happen if we stay together for another two years and then maybe he can get a better transfer."
The picture is less clear where Stekelenburg is concerned.
The 28-year-old will be out of contract next summer and De Boer is keen to quickly sort out the Holland international's future one way or the other.
"There is currently no interest in him," the coach said.
"We want to clarify whether he is staying or not and whether he wants to extend his current contract. If not, then a winter transfer seems probable."