The centre-half, 20, has been one of Town's star performers in his breakthrough season at the Broadfield.
Manager Richie Barker told the club website: "We are in discussions with a few players and they are all progressing smoothly but with regard to Joe we are very close and should be able to confirm something in the next few days.
"Joe has really enjoyed his first season at Crawley and feels he has progressed since coming here in August. He's quite a quiet lad but I have to say for a 20-year-old there has been a real authority and composure to his performances this season.
"I don't think Joe expected to play as many games as he has but he has come on tremendously since he came to the club and he has been one of our most consistent players this season. I'm sure when the fans award their player of the season he will be in contention.
"He has also enhanced his international career by playing for Wales Under-21s twice this season and I know he is very highly regarded by the coaches in the Welsh set-up."