Moore, whose second-half strike sealed Alex's 2-0 League Two play-off final victory over Cheltenham at Wembley last month, is out of contract but is under-23 so would command a compensation fee if he chooses to leave.
Railwaymen boss Steve Davis told his club's official website: "We have had early talks with Byron's agent. We very much want him to stay and be part of our plans here.
"I spoke with Byron three months ago telling him that I wanted him to stay and we have made him a good offer. We may have to tweak one or two things, like most contracts, but then that would be our final offer. We have done all we can and want to keep him.
"In my opinion he should remain with us and progress in League One. I think he will only enhance his reputation by staying here and showing what he can do at the higher level. He had his best season at the football club last time and 10 goals was a good return for a winger.
"We would like Byron to remain patient and trust our judgment because I think the bigger clubs would have come in for him. We can advise him and we have."