Stones said: "Jake and I have spoken twice this week in my office, and I have not changed my view that I would like him to stay on in one way or another in SA Rugby.
"He has asked for some time and this has been given to him. We would certainly like him to stay one way or another.
"But if he chooses to go overseas to pursue other options, we will wish him good luck."
White revealed last week he would not be renewing his contract with SA Rugby which expires on December 31.
He initially informed SA Rugby he did not intend to extend his spell in the head coach role, leading the governing body to advertise his position.
And even though he then hinted at a change of heart, SA Rugby have opted to choose from those who applied initially, a move White said "disappointed him" as he believes he should not have to reapply given his success.