Speed, who won 85 caps for his country, is set to be unveiled as John Toshack's successor on Tuesday afternoon, just four months after taking over at the Blades following Kevin Blackwell's departure.
And Edwards, who had two prolific spells with the club, believes Speed should be staying in South Yorkshire.
"I think his heart has ruled his head," Edwards told BBC Radio Sheffield. "In reality the job is trying to qualify for a competition, getting knocked out and going home early and then doing it all again next time round.
"They don't win and that's why it shouldn't appeal to him. And the really big challenge is to get this club back into play-off contention.
"United should have said no to the Welsh FA and told Speed he's got a job to do here. They gave him a three-year contract and that's maybe a decision they should have stuck by."
Source: PA
Source: PA