They announced the moves before Wednesday's deadline, also declining to offer arbitration to catcher Jason Varitek, right-handed knuckleballer Tim Wakefield, left-handed pitchers Erik Bedard and Trever Miller and outfielders J.D. Drew and Conor Jackson.
Ortiz, 36, has played for the Red Sox since 2003 while Wheeler finished his first season with the team.
The Sox are eligible to receive compensation for Ortiz, a Type A free agent, and Wheeler, a Type B free agent. The players have until December 7 to accept or decline that offer.
Varitek and Wakefield were cornerstones of the team that started a new winning tradition for the Red Sox, but both are on the downside of their careers.
Varitek, 39, has played all 15 of his major league seasons in Boston and Wakefield has pitched there since 1995, winning his 200th career game this past season.
Former Red Sox closer Jonathan Papelbon, who signed with Philadelphia earlier this month, was a Type A free agent and will yield Boston compensation.