The deal would be a two-year contract worth $9 to $10 million, a person familiar with the negotiations told The Associated Press.
Davis, 33, has spent the last three seasons with the Toronto Blue Jays, recording 125 stolen bases over that span. The eight-year veteran batted .260 with 45 steals in 108 games this past season.
Davis would join a Tigers outfield that features center fielder Austin Jackson and veteran right fielder Torii Hunter. Andy Dirks was Detroit's primary left fielder last season, batting .256 with nine home runs.
Detroit already added some speed to the lineup this offseason with the acquisition of second baseman Ian Kinsler, who was traded to the Tigers in the blockbuster deal that sent Prince Fielder to the Texas Rangers.
The Associated Press contributed to this report.