He returned to Ibrox on loan in January and the 34-year-old is a free agent in the summer.
Davis said: "We have had conversations and that's obviously something we will look to do. I am just concentrating on the games at the minute but I would like to be here.
"I feel I have got a big part to play, I feel I can bring a lot to the team. I feel we have a hungry group of players who have a real desire to start bringing trophies back to the club.
"There's been a lot of good work and good progress done this season and we need to try and take that to another level next year."