Narveson, who was on Miami's opening-day roster last season, will be at spring training with the AL champions this week in Goodyear, Arizona. The 35-year-old has a 30-19 major league record in 118 games (65 starts).
He has also pitched with St. Louis, Milwaukee. Narveson won a combined 23 games for the Brewers in 2010 and 2011.
Narveson will compete for one of the openings in Cleveland's bullpen. The Indians are looking to add another lefty to go with All-Star Andrew Miller and Boone Logan, who signed a one-year, $6.5 million deal with the club last week.
Narveson limited left-handed batters to a .218 average at Triple-A New Orleans and posted a 3.60 ERA in 15 starts.