The 26-year-old, who spent the second half of last season on loan at Sunderland, is currently out of favour at West Ham and was not in the squad for Saturday's opening game at Wolves.
He has been linked with a £1.5million move to Forest, although reports suggest Davies will initially try to sign the centre-half on loan.
Davies is still keen to strengthen his squad before the close of the transfer window and admits Davenport is among the Championship club's targets.
"It is a possibility," Davies told the Nottingham Evening Post. "Davenport is a name we have thought about.
"I am sitting down with my staff and putting together names to put to the board to recommend we should bring in.
"Let me say again, we will make sure that we are getting good value and that we are getting the right type of player.
"We are not going to bring players in for the sake of it."