Ramsey failed to agree terms on a new deal with the Gunners before his contract ran into its last six months, which allowed the midfielder to speak to clubs who expressed interest in signing the 28-year-old.
The pay will add up to as much as £83 million over four years, according to United Kingdom media reports.
As his contract will have ended, Arsenal will not be entitled to a transfer fee.
Welsh global Ramsey has already passed a January medical at Juve, and will link up with Massimiliano Allegri's team over the summer.
He has played more than 250 times for Arsenal, scoring 61 goals.
With Arsenal, Ramsey won three FA Cup titles and two Community Shields.