Although no official figures were confirmed on Monday, the commercial deal has been described as "the biggest deal in Puma and Arsenal's history".
The Gunners are understood to receive around £30million a year from main sponsors Emirates Airlines, and it has been reported the new deal will be for a similar amount when Puma takes over from Nike, who Arsenal have partnered since 1994.
Arsenal chief executive Ivan Gazidis said: "We are excited to be partnering with Puma, a company whose football heritage and record of innovation have a strong affinity with our own. This represents another important step forward in Arsenal's progression on and off the pitch."
Puma also has tie-ups with the Italian football federation and Borussia Dortmund, as well as current Arsenal forward Olivier Giroud and former Gunners captain Cesc Fabregas, now at Barcelona.
Puma's chief executive Bjoern Gulden added: "Arsenal have been a key strategic target for Puma for a number of years now.
"Through a clear commercial vision, a well-defined sports marketing strategy and a relentless enthusiasm within the Puma organisation, we are proud to have signed this partnership with a truly global football club.
"As we enter a new era in our company history, Arsenal represents a major commercial and marketing opportunity to reinforce Puma's credibility as a global sports brand, and we have full confidence the plans in place to activate this partnership will have a significant global impact."