Ashworth will leave Albion in June to take up a similar role with the FA.
The Daily Mail says Albion want to give the post to highly rated Warburton, who become sporting director at Brentford in 2011. He is understood to be making his mind up about the offer this week.
Warburton, a former City trader and academy manager at Watford, is best known for being the co-founder of the NextGen series, a burgeoning European tournament for Under 19s heralded as the Champions League for youngsters.
And it will be Warburton's vision for young players that will have attracted Albion, who have lost Roy Hodgson and Ashworth to the FA in the last year.