Collymore wrote in his column for the People: "Martin O'Neill is not a quitter and can still turn around Sunderland's season - but he will need financial backing in January to do so.
"Last week, there were rumours he had left his job due to the club's position towards the foot of the table and a bad run of form.
"And I wrote on Twitter that there were unconfirmed reports he had resigned.
"Clearly, it was not true, and a few days ago I spoke to him over the phone to apologise for my part in spreading the rumour that had been passed on to me.
"I am delighted I was wrong too, because there is a massive job to be done at the club and I have every belief that O'Neill is the only man to do it.
"I played for him at Leicester City and know exactly what he is capable of in terms of motivation - and he has a great staff, with people like Steve Walford, Seamus McDonagh and Steve Guppy behind him.
"So there's nothing wrong with the back room set-up at the club."