The Black Cats boss is adamant that reported interest in the 27-year-old midfielder is little more than hear-say at the moment, and having spoken to the club's executives can confirm that no offer has been received.
"Arsenal have not been in touch with me, I don't know anything about it," said O'Neill. "I asked the chief executive as well and she said, 'no, there hasn't been anybody in touch yet'.
"But what I have said is that I would keep players informed about what is happening.
"Obviously, I don't want to rush players to leave the football club. I would rather think about stabilising things here, and hopefully we can try to progress from there.
"But regardless of whether you are going to be doing business or not, you are going to be linked with players, clubs are going to be linked with some of our players, and at this minute we are hanging in there."
Wenger has major concerns in the left-back position, with Kieran Gibbs, Andre Santos and Thomas Vermaelen all currently out injured for Arsenal, so Richardson could be signed as short-term cover in that area.