Racing Metro are leading the chase for the Blues star and have reportedly tabled a £330,000-a-year offer.
Cardiff Blues have also made an offered Cuthbert a new contract and are getting increasingly frustrated with the 22-year-old's indecision.
But former Wales lockl Davies reckons Cuthbert should stick with the Blues to safeguard his long-term development.
"I want him to stay and I think he should stay," said Davies.
"Maybe I'm a bit biased but he has come on the scene, he's a young player who has been well-coached and his performances have been superb.
"But he is still learning the game and developing as a professional. That is important.
"Being a professional is about playing on the field but there is an immense amount of hard work that goes into that.
"The region have dealt with it very professionally, it's amicable and it's a case of waiting for the player to make a decision.
"Being biased and understanding the development process, I feel he has a very strong support network in Wales. Maybe that will be the same in France, but we all talk about second season syndrome.
"For me he is in a growing phase and has to think this through carefully. He has a long time left in the game.
"He is still learning his trade but whatever decision he makes I will respect and wish him well.
"We want him to stay but it is up to him. Hopefully, we will have a decision shortly."