The 21-year-old came through the Shakers' youth system and helped Macclesfield secure promotion to League Two during a loan spell in 2018.
Burgess spent last season on loan at Wrexham and then York.
Vale boss John Askey said on the club's official website: "He's a player that I've had before and he's a good lad. He's got the potential to break into the first team so hopefully, he can do that.
"We're giving him a chance to show that he's got what it takes to perform in a Vale shirt and find the form he's showed at previous clubs. Hopefully he can make the breakthrough."