Colclough’s current deal expires this summer, but the 20-year-old is expected to commit to the Railwaymen.
Davis told the club’s official website: “The club has made Ryan an offer and I don’t envisage too many problems there.
“I know his agent came into the club (on Monday) and hopefully we are close with that.
“I think Ryan appreciates that the club stood by him in some difficult times and he knows that this is the best place for him to develop.
“He came back from a serious injury and played his part in the run-in. We know he can be a potential match-winner for us.
“Ryan needs to play games and he has a good support network here that would allow him to play, develop and get him where he should be in the future.”
Colclough returned from a 16-month pelvic injury lay-off for the club’s successful run-in, scoring in away wins at Sheffield United and Coventry as Crewe finished two points clear of the relegation zone.
Davis is also hoping midfielder Anthony Grant opts to sign a new deal as he too would be available on a free transfer later this summer.
The Alex manager added: “I will talk with him again to see if he would consider us as one of his options going forward. He is a London boy but I am sure he enjoyed his time here.
“He is 27 now and I am not aware that he wants to return south. All we can do is when the time is right is sit down with him and declare our interest in him staying here.”