Buxton was speaking on Channel 4 News in a debate about Thompson's confirmation that 6 Music and Asian Network are facing closure at the end of 2011.
Defending his employer, the comedian claimed that 6 Music was a station that created shows "you couldn't find anywhere else".
"As far as our show is concerned, and a lot on 6 Music, we have got an incredibly devoted and intelligent and unusually witty and creative audience. They aren't getting those shows anywhere else and I don't think they would if 6 Music didn't exist," he said.
When he was asked what he would like to ask Thompson, Buxton replied: "Mark 'Thomo' Thompson. I am inviting you out for a fight. I could take you. I have been practising.
"No, seriously, I want to know if it is a complete fait accompli or whether the BBC will listen to people's opinions if they're offered to the BBC Trust online.
"Part two - of my two part question - I don't understand why it is necessary to get rid of a station in the name of quality when it is a station that is working really well and is really loved. It is working because it is part of the BBC and it seems to be doing everything that the BBC wants to do best."
> Trust: 'Protests could save BBC 6 Music'