Kenny played just twice for the Blades this season after serving a nine-month ban imposed by the Football Association for failing a drugs test.
A move to QPR would reunite Kenny with manager Neil Warnock who signed him for the Blades from Bury in 2002.
The bid is thought to be some way short of Sheffield United's valuation of the player.
They plan to hold talks with Kenny in the next few days in a bid to keep him at Bramall Lane.
Source: BBC Sport
Source: BBC Sport