Moore is out of contract this summer, but Crewe boss Steve Davis is hoping the 23-year-old will sign a new deal now they are in League One.
"We're in the very early stages of talks with Byron and his representative and remain hopeful that he will choose to stay with us," Davis told the club's official website.
"We want Byron to remain patient and stay with the group because the move will follow. We want him to improve and enhance his reputation with us.
"We would receive a fee for him because of his age, if he was to remain in this country. Of course he could move to a foreign club or to Scotland to escape that, but we can advise him on what we think is best for his career.
Moore, who began his career as a trainee at Gresty Road, scored 10 goals in 49 appearances last season, including the second to seal his side's 2-0 win over Cheltenham in last month's play-off final.
