The Ibrox club have been chasing Davis' signature since his successful loan spell ended at the end of last season.
But Gers boss Walter Smith has not been able to complete the signing, despite his desperate need for an inventive midfielder after Barry Ferguson's injury.
And Hodgson has now hinted that the SPL side could be set to miss out on the 23-year-old altogether, telling the Daily Record: "They say Rangers have not had much money until now.
"But I find that hard to understand because they get 50,000 people every other week and it is a very big club.
"We are being obliged to contemplate transfers of £6m, £8m and that type of figure, so it seems strange a club of that stature can only afford what they have offered us.
"I am perfectly happy to keep Steven. He is a good player and I would be very disappointed if the club allowed him to go for a sum of money that is below his worth."



