Monbiot is among a number of individuals facing potential legal action after wrongly naming the former Tory Treasurer on Twitter in connection with the North Wales child sex abuse scandal.
This followed a BBC documentary on the abuse that linked a "senior Tory politician" but did not name Lord McAlpine.
Monbiot will have to pay his own legal fees and any compensation costs if he has to reach a settlement with McAlpine, as Guardian News & Media has said that he tweeted from his own personal account.
Monbiot has already issued a full and unreserved apology over his tweet to 56,000 followers, which said: "I looked up Lord #McAlpine on t'internet. It says the strangest things. I can confirm that Lord #McAlpine was Conservative Party Treasurer when Mrs Thatcher was prime minister."
He has since written to Lord McAlpine twice to apologise and posted a public apology in a later blog post.
"The tweets I sent which hinted - as I assumed to be the case - that Lord McAlpine was the person the child abuse victim Steve Messham was talking about were so idiotic that, looking back on them today, I cannot believe that I wrote them. But I did, and they are unforgivable.
"I helped to stoke an atmosphere of febrile innuendo around an innocent man, and I am desperately sorry for the harm I have done him.
"I have set out, throughout my adult life, to try to do good; instead I have now played a part in inflicting a terrible hurt upon someone who had done none of the harm of which he was wrongly accused. I apologise abjectly and unreservedly to Lord McAlpine."
But it is not guaranteed that legal action will not follow, as Lord McAlpine has indicated that he will sue anyone who mentioned his name in connection with the false rumours.
> Lord McAlpine on 'Newsnight' scandal: 'The BBC should have called me'
His lawyer, who has already secured £185,000 in compensation from the BBC, has warned that "trial by Twitter must end".
Guardian News & Media, confirming that it would not pay the columnist's fees should legal action follow, has today said: "George Monbiot is not a member of the Guardian staff and was tweeting in a personal capacity.
"George has acknowledged that his tweets were wholly inappropriate and a serious error and has posted a comprehensive apology on his own blog.
"The Guardian cannot and will not take responsibility for comments that were made by a contributor in a personal capacity, however we have taken this opportunity to remind our journalists and freelancers, including George, that they should apply the same ethical and legal standards to what they say on social media as they do to what they write for The Guardian."
The paper said that it would continue to publish Monbiot's column "for the foreseeable future".
Meanwhile, Sally Bercow has denied that she sent a libellous tweet about Lord McAlpine, and instead claimed she was just enquiring why his name was trending on the site.