Olympic gold medallist and reigning Tour de France champion Wiggins has recorded a two-hour programme with Weller which will be broadcast on Boxing Day (Wednesday, December 26).
The six-minute preview features highlights from the show and includes Wiggins talking to Weller about his London 2012 and Tour de France experiences.
The cyclist also explains his passion for mod culture, saying that going to see Ocean Colour Scene play live at the Shepherd's Bush Empire made him feel like he was "part of something... part of a group".
Wiggins plays down rumours that he could be knighted in the New Years Honours list during the preview, arguing that Mo Farah and Ben Ainslie were deserving of such an accolade.
"Apparently they're only allowed to give two out a year... I'm probably last on the list," he jokes.
Wiggins was named the BBC Sports Personality of the Year for 2012 on Sunday (December 16), ahead of Olympic champion heptathlete Jessica Ennis, plus London 2012 gold medallist US Open champion Andy Murray.
His victory was seen by 14.5 million viewers, compared to a peak audience of only 4.2 million for last year's ceremony.
When Bradley Wiggins Met Paul Weller airs on Boxing Day from 1pm to 3pm on BBC 6Music.
Photo gallery - BBC Sports Personality of the Year 2012:
Copyright: PA Images David Davies/PA Wire