Speed, who represented Wales 85 times, had been strongly tipped to become the successor to Paulo Sousa at the Liberty Stadium.
But the Blades are desperate to keep their highly rated coach at Bramall Lane, despite Speed himself reportedly showing an interest in taking the Swans job.
Chairman Huw Jenkins has confirmed that after being turned down by United, the club will move onto other managerial targets.
"As was the case with a few managers, an approach was made for Gary.
"It was turned down though and, once that happens, there is nothing we can do about it.
"That’s why you have several options. It’s not a disappointment as such because I have done this job long enough to know that you can’t just focus on one target.
"Sometimes they don’t come off and you accept that no means no.
"In terms of Gary, that brings an end to it."
This article first appeared on JACKARMY.net.