The businessman is due to leave The Hawthorns at the end of the season to become the Football Association's director of elite development.
"There is no time pressure on this, we don't need to make any knee-jerk decisions," said Ashworth.
"A lot of people have applied for the role. We've had approaches from abroad, especially as the role is more common in Europe, but we don't have a timeframe on a shortlist. We are very much going through the process.
"We're still sifting through applications, talking to people and preparing a shortlist to ultimately make sure we get the right person."
Ashworth spent five years working with the Baggies.