The Baggies gave the Scot his first manager's job in the summer of 2012 after he had spent years working as an assistant at several clubs, but he lasted only 18 months before being fired at the weekend with the Baggies 16th in the table, just two points above the relegation zone.
Atkinson believes this is not the time to be experimenting again and would like to see them bring in someone with a track record in the English top flight.
"Steve Clarke came in at the start of a season but this situation is totally different," Atkinson told the Express and Star.
"They have decided they need to replace him so they now need to look for somebody with experience in the Premier League, but who they go for I don't know because there isn't anyone obvious that jumps out.
"If you were to ask me, I couldn't really name anyone.
"A few weeks ago, maybe Steve McClaren would have been an obvious candidate but he's got a job now.
"Alex McLeish still lives locally and has a fair bit of experience of fighting fires but other than that nobody springs to mind.
"It is really tough to say who they will go for now.
"Unless they've already got someone lined up, which they don't appear to have, I have no idea who they will pick."
A whole host of potential candidates have been mooted so far and include Red Bull Salzburg sporting director Ralf Rangnick, former Manchester United striker and current Molde coach Ole Gunnar Solskjaer.
Mike Phelan, Sir Alex Ferguson's long-time assistant at Old Trafford, Cardiff boss Malky McKay, recently-resigned Watford boss Gianfranco Zola and sacked Fulham manager Martin Jol have also been linked with the role.