The giant wing, who raced to three tries as Wales clinched the Grand Slam, has been targeted by clubs in France.
Cardiff Blues have also made an offer for Cuthbert and Davies is convinced staying with the Welsh region is the safest option.
“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.”
The former Scarlets boss has undertaken a fact finding mission during his first days in charge of the Blues and will now reflect on the current structures within the region.
Gareth Baber and Justin Burnell, who led the Blues this season, have another year under contract and Davies expects them to remain a part of the coaching set-up.
“I’ve spent two days meeting people and have a fair idea about what people do and don’t do at the region,” said Davies.
“The next couple of weeks will be a period of reflection for me, but we have a lot of capable people here and we’ll see how that process moves forward in the summer.
“What I have done over the last couple of days is look at the structure itself and see who is in place and who is doing what.
“Obviously their roles are changing for next season. It is a case of going away and looking at things.
“There are a lot of careers being developed at the Blues. We have to try and get all the facts and look at it and make some considered decisions to take the region forward in the best possible way over the next few years.”
