The Buffalo Bills also tried to sign Smith, but he ultimately decided he wanted to return to Cincinnati, where he spent his first seven seasons, the source said.
Smith staked his claim to the Vikings' starting right tackle job last season after lingering Achilles issues forced Phil Loadholt to retire in July. Smith played just three games before undergoing season-ending surgery to repair a torn triceps muscle.
An eight-year veteran, Smith had signed a one-year, $3.5 million deal last March with the Vikings, reuniting him with head coach Mike Zimmer, who was Cincinnati's defensive coordinator for Smith's first five seasons there.
Smith, who turned 30 in January, struggled at the beginning of the season as the Vikings sought to improve an offensive line that had been their biggest weakness in 2015. Injuries to Smith and Matt Kalil, however, had the Vikings shuttling through tackle options for the rest of the season, and their line once again became their biggest problem on offense.
Smith has made 77 career starts in his eight NFL seasons.
ESPN's Ben Goessling contributed to this report.