Bailey is considered one of the top talents still available in free agency having been released by the Broncos after an injury-hit 2013 season.
The 35-year-old was due to count $10m (£6m) against the Broncos' salary cap next season, prompting the Super Bowl runners-up to part company with the veteran.
However, the 12-time Pro Bowler, who is widely regarded as one of the best cornerbacks in NFL history, has made it clear that he is not planning on retiring and wants to play again in the 2014 season.
Bailey is reportedly keen to visit the Saints' facility at least one more time before making a decision, depending on how talks progress.
The former first-round Draft pick has recorded 52 interceptions in his illustrious 15-year career with the Broncos and the Washington Redskins.