Clarke has made a promising start to his managerial career, guiding the Baggies to sixth place in the Premier League after eight games.
But while Ashworth is delighted with the progress the current coach is making, he has stressed the need for forward-thinking in case Clarke ever decides to leave.
"Steve might get poached by someone. That's the reality we live in," Ashworth told reporters.
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"I remember reading a piece about an interview I did in the programme just after Robbie [Di Matteo] started where I said my job now was to look for the next head coach and it came across as 'how disrespectful'. But it isn't.
"My job is always to drive forward and help find the next coach."
Ashworth is due to leave West Brom to take on his new role as FA director of elite development next summer, but the 41-year-old admits that his only concern right now is how Clarke fares in a role Ashworth personally selected him for.
He added: "I spent a lot of time with Steve during the interviewing process. I remember going to see him in Peterborough on a fishing trip. It is in my interests for Steve to succeed.
"Because if a head coach fails it's like having a failed player – it's an appointment you, along with the rest of the board, have made. And if it fails it's on record.
"I have to try to make him successful."