West Brom today announced Pulis has been axed after a run of 11 games without a win.
Saturday’s 4-0 loss at home to Chelsea was their fourth defeat on the bounce.
He leaves the Baggies 17th in the table and a point off the relegation zone.
Former England manager Allardyce has been tipped for the role.
But Winterburn doubts the 63-year-old would go for the hawthorns post.
“Listening to Sam, I think he would need a job that would excite him. He would maybe want to take over as an international manager,” former Arsenal defender Winterburn said on Sky Sports.
“Would West Brom excite him? I’m not sure, to be honest. Or would it need to be a club where he feels he deserves a chance with.
“A team he thinks can sit comfortably in the top half of the league and be pushing for a European place.
“I just get the feeling the way he talks at times that he would be open to an international job but wouldn’t want to take over a team that is perceived to be struggling.”
West Brom join Everton as clubs seeking a new manager, with the Toffees having been unsuccessful in their search of replacing Ronald Koeman.
And while Winterburn reckons Pulis will get the Wales post, he is concerned the Baggies will struggle if they don’t install a new boss in good time to prepare for the January transfer window.
“We’re getting the same situation with Everton,” he said.
“They’ve sacked the manager without knowing who to bring in. As a club you have to have a plan, you have to have the manager there tat you think is right for your club and you’ve got to be pretty confident that you can bring him in.
“Because if you start scrapping around for four or five weeks, while you’re coming into a transfer window, you haven’t got a manager in charge long term who you can trust.
“Someone you can trust in the transfer window to alter the squad, getting in new talent to liven the squad up. Otherwise you’re drifting.
“Hopefully they have a plan to get a new manager to bring in and turn it around. Otherwise I just find the situation, particularly with Everton, fascinating. Let’s see how long it takes West Brom to put someone in place.”