Rogers' deal is worth $4 million, a league source told ESPN NFL insider Adam Schefter.
Rogers, a former three-time Pro Bowl player, was cut on Feb. 9 after playing sparingly during the 2010 season due to ankle, hip and elbow injuries. He had been scheduled to make $5.5 million in 2011 with a $500,000 roster bonus.
NFL free agency is scheduled to begin March 4 -- pending the outcome of negotiations for a new collective bargaining agreement -- but Rogers is free to sign with a new team now because the Browns terminated his contract early.
While Redskins and Chiefs wanted him as a nose tackle for their 3-4 defenses, Rogers preferred the Saints for two reasons. The one-year deal will allow him to be a free agent after the season. Also, the Saints defense will play him in a 4-3 alignment, although defensive coordinator Gregg Williams switches into a three-man line often.
The Saints also reached an agreement with kicker Garrett Hartley on a five-year deal, a league source told Schefter. Hartley is scheduled to sign the deal Wednesday.
Saints general manager Mickey Loomis says the club on Tuesday also tendered offers to 10 restricted free agents. However, those tenders could be voided by a new collective bargaining agreement currently being negotiated by NFL owners and the players' union.
The tenders were offered to defensive tackle Remi Ayodele, left tackle Jermon Bushrod, left guard Carl Nicks, safeties Roman Harper and Usama Young, running back Pierre Thomas, receiver Lance Moore, tight end David Thomas, Hartley and defensive end Jeff Charleston.
Information from ESPN.com's senior NFL writer John Clayton and The Associated Press contributed to this report.