The BBC Sport pundit believes his fellow Welshman will return to the Premier League soon.
The Welshman has been out of work since leaving Crystal Palace earlier this season but Harry Redknapp, Nigel Pearson and Alan Irvine are all under pressure at their current clubs.
And the former Premier League midfielder thinks Pulis's success at Palace will put the trio under extra pressure while he is available.
"Forget unhappy fans, player power or impatient owners - there is one big reason why managers of teams at the bottom of the table are at risk and his name is Tony Pulis," Savage wrote in his weekly BBC Sport column.
"It is not his fault but all the chairmen and owners out there will see what Pulis did at Crystal Palace last season, when he came in and kept the Eagles up, and think 'he saved them and he can save us'.
Pulis managed Stoke City in the Premier League before his later spell with Crystal Palace, when he kept them up comfortably under difficult circumstances last campaign.