The 24-year-old 6`3” Archer joins the Shakers on a two-year deal after the clubs agreed an undisclosed transfer fee.
Archer joined Chester last October and scored 4 goals in 11 starts after impressing for Evo-Stik Northern Premier League outfit Stourbridge, where he scored 22 goals in 51 appearances.
Speaking to https://www.buryfc.co.uk, manager Ryan Lowe said: “I'm really pleased to sign Jordan as he was a wanted man and we have managed to get the deal done for him.
“A lot of credit has to go to Lee Dykes who has monitored Jordan for a while. Once we put everything together, we knew that Jordan fit the bill of what we wanted.
“He is a powerful striker who runs in behind, and he is hungry to succeed. His hunger and desire are what gave me a good feel for him.
“It will certainly take Jordan a little time to adapt to full-time football, but I’m happy to put the work in with him if it's going to pay off, and I’m sure it will!
“Big thanks has to go to the Chairman again for backing me with my recruitment, especially with us paying a fee.
Archer spoke of his delight at signing, saying: “I'm delighted to be joining Bury. When I heard the club were interested, I made my decision to join straight away without hesitation.
“It's a step up in my career; I can't wait to get going and repay the club for bringing me to the Football League.
“I’m confident I can make the step up to the League. I'll hopefully repay the club's faith by scoring goals and working hard for the team and manager”.