The 27-year-old is on trial at Elland Road having missed all of last season after suffering serious leg injuries in an off-the-pitch incident.
Davenport has been without a club since his contract with West Ham was terminated five months ago but Grayson is hopeful a deal can be concluded.
"He's had a long time out of football - 12 months out with a serious injury that's stopping him from playing at all," Grayson told the Yorkshire Evening Post.
"The deal won't be concluded in the next week or two because he's quite a long way from being fit, but we're trying to get him fit.
"If everything goes well then I think we've got a player on our hands who can get back to the level he was at with a top club like West Ham."
Source: PA
Source: PA