Ashworth is leaving his job as Albion's sporting and technical director to take up a similar role with the Football Association at the new National Football Centre at St George's Park next summer, but he is playing a key part in identifying his replacement at The Hawthorns.
Baggies chairman Jeremy Peace would like to make an appointment sooner rather than later so they can work alongside Ashworth before he switches to his new position.
Warburton had previously had a post with Watford's academy and was a coach at Brentford before being promoted to his current role.
Albion know the importance of finding the right replacement for Ashworth, who was responsible for discovering the likes of striker Peter Odemwingie and midfielders Youssouf Mulumbu and Claudio Yacob.
They had earmarked Hamburg sporting director Lee Congleton, who previously worked at Chelsea, but were denied permission to speak to him by the Bundesliga club.
Albion have also been linked with ex-Liverpool and West Ham sporting director Damien Comolli.