Gillingham boss Ronnie Jepson had earlier revealed that the left-back, 24, would be leaving Priestfield.
Jackman previously worked with Cobblers boss Stuart Gray at Aston Villa.
"He knows how I work and I know how Danny works. He has been a big influence at both Stockport and Gillingham," Gray told Town's website.
Jepson told BBC Radio Kent: "He proved a few points to me this season and by the end of the campaign he had certainly improved in a few areas.
"It will be sad to see him go, but all we're interested in is people who want to play at this football club."
BBC