Saints boss Burley has been confirmed as the man the SFA want to succeed Alex McLeish as the new national team boss.
A spokesman for the SFA has said: "We have had initial discussions with the manager's agent and these will continue tomorrow.
"But we have reached agreement with Southampton."
Scottish Football Association president George Peat - who was on the interview panel - stopped short of naming the 51-year-old as he arrived for the board meeting at Hampden Park, but told Sky Sports News: "We think we've got the right man."
He added: "We've gone down the same route as we did with the appointments of Walter Smith and Alex McLeish.
"It's been difficult from the point of view there's been little to choose between the candidates we were discussing. But we think we've got the right man."