Garcon, who has deep speed, has been in talks with the Colts all weekend and again on Monday morning. Though he plans to stay in touch with the Colts, the source said, Garcon felt the offers weren't good enough to keep him off the free-agent market.
The Colts have until 4 p.m. ET Tuesday to reach a deal with their unsigned free agents before they are free to sign with other teams.
Garcon had a career year in 2011 even though quarterback Peyton Manning missed the entire season after neck surgery.
The 25-year-old receiver had 70 receptions for 947 yards and six touchdowns, catching passes from a carousel of quarterbacks.
Information from The Associated Press was used in this report.