Clough is a big admirer of White and is thought to have tried to sign him during his time in charge of Derby County.
White has found himself down the pecking order at Leeds this season, making just three appearances for Brian McDermott's side.
Clough is now waiting to see if Leeds are willing to loan out White as he looks to bolster his squad at Bramall Lane.
"Yes, we've asked about Aidan," Clough told the Sheffield Star.
"Nothing more has been done yet but hopefully we should know more around Tuesday or Wednesday. It's a possibility.
"He's a player that we know. That was about 18 months ago so we'll see what happens."