Among those also named as first-time finalists were three-time National Collegiate Athletic Association (NCAA) Coach of the Year Sylvia Hatchell, WNBA All-Star guard Dawn Staley and six-time NBA All-Star Mitch Richmond.
Payton, a point guard and defensive specialist for 18 NBA seasons, ranks in the top 10 all time in assists and steals.
"A guy coming from Oakland, California, I never dreamed of this," said Payton, who played 13 seasons with the Seattle SuperSonics and five with the Miami Heat before retiring after the 2006-07 campaign.
"The things I did at the NBA, I never dreamed of that. I would be honored to be a first ballot."
Previous finalists again being considered for induction into the Springfield, Massachusetts, hall are former NBA All-Stars Maurice Cheeks, Spencer Haywood, and Bernard King, and college coaches Guy Lewis, Rick Pitino and Jerry Tarkanian.
The Class of 2013 will be unveiled at the NCAA Final Four on April 8 and the enshrinement will take place Sept. 8.
Already a Hall of Famer as a player, Heinsohn is up for enshrinement as a coach, having led the Boston Celtics to titles in 1974 and 1976.
Hardaway, who had a stellar, 13-year NBA career with five teams, scored 5,000 points and had 2,500 assists faster than any other player, except Oscar Robertson.
The announcement of the finalists was made during the festivities leading up to the NBA All-Star game on Sunday.