The writer pursued Armstrong's use of drugs for 13 years but could never prove his claims. The paper was sued by Armstrong and in 2006 the newspaper settled the case for more than $500,000 (£315,000) after spending more than $1m (£630,000) in legal fees.
The cyclist recently confessed to Oprah Winfrey to doping throughout his career and the talkshow host named Walsh in particular during the second part of the interview broadcast last night.
In response, Armstrong said: "I would apologise to David."
Walsh, 52, has this morning responded on Twitter. He wrote: "Oprah pressured him, the apology was, I thought, hesitantly promised. I didn't ask for it, or expect it, but, yes, if ['the chance to ask the questions or receive all the apologies in the world is'] offered, I accept."
Walsh had said that the first half of the Armstrong interview on Thursday didn't go "far enough".
Click here to read our blow-by-blow account of yesterday's conclusion to the interview.