Reds boss Steve Evans told the club website: "He had a brilliant season and we have watched and analysed him for around six months.
"Tyrone is magnificent in the air and he gets all sorts of goals. That is the sign of a top-class striker.
Barnett added: "I enjoyed my time at Macclesfield but the time was right to move on.
"Steve Evans spoke to me in depth about what my role in the team would be but he made it clear I had to fight for my place.
"I don't mind that challenge. I cannot wait to get started."
Source: BBC Sport
Source: BBC Sport