Brailsford is the former performance director at British Cycling and is now director of sport at Ineos, founded by Sir Jim Ratcliffe.
It was confirmed on Christmas Eve that the British billionaire will purchase a 25 per cent stake in United worth around £1.25billion.
As part of his work with Ineos, Brailsford has worked at Ligue 1 side Nice and looks set to take a leading role at Old Trafford with Ratcliffe as he was spotted in the stands for the club's dramatic 3-2 win over Aston Villa.
The 59-year-old, who is also team principal of cycling UCI WorldTeam Ineos Grenadiers [formerly Team Sky], is an esteemed figure in cycling but Saunders has questioned if his experience can be transferable at Man United.
"It needs a complete reshuffle again but I wouldn't have a cycling expert doing it for me, I'd have an expert in football," he said on talkSPORT Breakfast.
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Host Jamie O'Hara asked who he was talking about and Saunders replied: "Dave Brailsford, who's the guru with cyclists in the Tour de France and the Olympics, and his motto is 'Can I get an extra one per cent out of everything?'
"Fair enough in cycling, but when you've got to find out who's lying to you in football, like agents and other clubs and players, everybody lies to you, so experience makes you listen to the right people.
"You cannot come into football and think because it worked in cycling you're going to takeover a football club and get inside a footballer's mind. They're not cyclists, it's not an individual sport, it's a team sport.
"You try and get in [Jadon] Sancho's mind! You think Dave Brailsford is going to get inside Sancho's mind and get one per cent out of him? The only thing he's going to make him do is play better at PlayStation."
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Amid the incoming £236million worth of investment, Saunders added: "They're about to make another mistake with a load of money again, I don't know what part he's going to play."
O'Hara told him to give Brailsford a chance, but Saunders insisted: "They gave Ed Woodward a chance and the recruitment was... recruitment is everything at a football club.
Post-match press conference: Erik ten Hag (Premier League: Manchester United 3-2 Aston Villa)
"You've got to know what you're doing. You've got to know how it works on the inside and outside, you've got to look at a player and think, 'You'll do for us.'"
Ratcliffe and his Ineos team have been handed responsibility of the football operations at United, who sit sixth in the Premier League.
The Glazer family remain the club's controlling shareholder after they revealed towards the end of 2022 they were open to a sale.
Qatar's Sheikh Jassim bin Hamad Al Thani competed with Ratcliffe for a takeover but the former pulled out of the deal and allowed the latter to proceed with his investment.