Former Sheffield United boss Warnock is on a five-man shortlist to replace Dave Jones, who was axed following last weekend's 2-0 defeat to Blackpool at Bloomfield Road.
Beagrie thinks it would be brave of Hillsborough chairman Mandaric to hire a man so intertwined with Wednesday's local rivals - but reckons Warnock could aid the Yorkshiremen's survival bid if he was handed the managerial reins.
"Neil is like Marmite and fans either love him or hate him, but he knows the Championship inside out and so will be mentioned for jobs in the division," Beagrie told FL72 - Preview.
"Will Milan, who has an ego as big as Neil's, be willing to work with him? It could be a success but do you want to start on a negative as far as 50 per cent of the supporters are concerned? I don't know."
Caretaker Wednesday boss Stuart Gray guided his charges to a shock 2-1 win over Championship leaders Leicester - and to within three points of safety - on Tuesday evening, with on-loan Sunderland man Connor Wickham netting a brace.
The 20-year-old has struggled to impose himself at the Stadium of Light since signing from Ipswich for £8million in June 2011, but Beagrie says he is revelling in being Wednesday's go-to man.
"Wickham has got real quality," said the ex-Sheffield United winger. "He has got good feet for a big fella, tremendous skill levels and is quite mobile as well.
"Wednesday are building their team around him and he is going to be pivotal to their success this year."
Beagrie's former club Middlesbrough, meanwhile, have lost two of the three games in which they have been bossed by new manager Aitor Karanka following Wednesday's last-ditch defeat at play-off chasing Derby.
However, the Sky Sports pundit says Boro supporters have been excited by the arrival of the ex-Real Madrid and Athletic Bilbao defender, and thinks Karanka could thrive on Teesside if he is able to get the best out of his two talented wingers.
"Middlesbrough fans are delighted that the club has thought outside the box and Karanka has some good players there to work with," added Beagrie, who made 33 league appearances for the North East club in the 1980s.
"Albert Adomah and Mustapha Carayol, in particular, are big threats in the wide areas, but Boro may have to play a more counter-attacking style to get towards the top eight, which is where everyone anticipates them being."
Catch all the goals from this weekend's Championship fixtures in FL72-Review, 6pm, Monday, Sky Sports 1.