Speed has been given permission by Sheffield United to speak to the Welsh FA with regard to the position.
It is understood the Welsh FA are likely to confirm Speed in the post on Tuesday.
A statement from Sheffield United: "Sheffield United have given Gary Speed permission to speak to the Welsh FA with regard to the position of international manager, and a further statement is expected to be made on Tuesday.
"Meanwhile, coach John Carver will continue to take the Blades' training session as normal."
Speed's almost certain exit has brought about rumours that Chris Hughton could be handed the job.