His current deal expires after next year's World Cup but his request for an extension until after the British & Irish Lions tour in 2009 has been turned down.
But White has confirmed he would lead the side in the Tri-Nations series after a "fruitful" meeting with Hoskins and Tshume, who confirmed the president's council will make a decision on the matter next week.
"I am very happy with the genuine approach and the sense of sincerity they have displayed around the issue of my contract," said White.
"I want to state clearly that it remains a great honour and privilege to coach the Springboks. I am loyal to the national team.
"During our meeting, they showed me they understand my reasons for raising the matter of the contract, and that I never wanted to leave the country. It was all a matter of seeking common understanding on future plans for all the parties involved in our rugby, particularly the national team.
"It is important for the team to know that when we assemble in Johannesburg next week, there are no uncertainties about me and my role.
"Our focus as a team remains the same. We are preparing for a tour that is going to be tough. We therefore need to be completely focused when we assemble.
"I do not have control over what the president's council decides when it meets. That is out of my hands and I will have to respect the decision they reach."
A joint statement by Hoskins and Tshume confirmed it had been a "constructive" meeting.
"We had a positive and constructive meeting with Jake, which helped clear the air after a wide range of speculation in recent media reports," read the statement.
"We are pleased to say that the coach and the leadership of SA Rugby are totally aligned on the process to be followed regarding the issue of the extension of his contract. He respects the fact that the president's council will deliberate on the matter and come up with a decision.
"It is important to note that the meeting of the president's council next Monday and Tuesday was scheduled long before the issue of the coach's contract was discussed publicly. The matter of the coach's and players' contracts is a subject that constantly arises as part of the broader discussions about the future of rugby in South Africa.
"Whether the president's council will immediately provide us with the final decision on this matter after its deliberations depends entirely on the council."
The Springboks squad meet on Monday ahead of two Tests against Australia and one against New Zealand on their five-week trip.