The disgraced US rider, stripped of his seven Tour de France titles and banned for life for serial drug-taking, held face-to-face talks with Bassons in a Paris hotel, sports daily L'Equipe reported .
"The most important thing for me is to tell you in private that I'm sorry," the paper quoted Armstrong as telling Bassons in the meeting.
Bassons was riding for Festina when the team became embroiled in a doping scandal in 1998.
The following year he abandoned his one and only Tour de France after a posse of riders, led by Armstrong, made their dissatisfaction clear following his anti-doping sentiments expressed in a newspaper column.
He retired from professional cycling in 2001.
This latest apology follows Armstrong's meeting last month with Emma O'Reilly, the whistleblowing masseuse on Armstrong's former US Postal team.
According to L'Equipe Armstrong is on a kind of "tour of redemption".
Before arriving in Paris he was in Rome to pave the way for an eventual meeting with old foe Filippo Simeoni, the former Italian cyclist he threatened in 2004.