The veteran broadcaster and former The Sunday Times editor quizzed his more than 300,000 Twitter followers on whether the rumours about the two Commons colleagues was true.
Abbott now sits in Corbyn's shadow cabinet as Shadow International Development Secretary.
But Ruth Davidson, leader of the Conservatives in Scotland and a prominent political tweeter, hit back at the question, pointing out that the "brief fling", as described by The Times, was purported to have happened before either was elected to Parliament.
The Times' story
"More importantly, who cares?" she hit back at Neil, adding: "As single adults before they were elected to Commons, surely this fails IPSO [the new press regulator] public interest test?"
ITV's Europe Editor James Mates quickly waded in, commenting that, given Corbyn this week promoted Abbott to his front bench team, the story stood.
His response prompted Telegraph columnist and Labour supporter Dan Hodges to chime in too, asking: "We really saying we need to [sic] full disclosure on past relationships every time there's a reshuffle?'
Neil responded to the criticism, telling 36-year-old Davidson she should report newspapers that ran with the piece to Britain's press regulator if she really thought it breached their ethics code.
"I'm sure Labour can fight their own battles. I get that there's scrutiny on potential PMs, but a 40y.o fling? Who's business is that?" she hit back.
Responding to Neil, who retorted that that issue was "trivial", Davidson warned that there could now be "plenty of politicians from their 20s" who had become "fair game" because of the reports carried by newspapers.
She added as an afterthought:
The news comes just one day after Corbyn took to the Commons to face Prime Minister David Cameron over the dispatch box, in his first verbal jousting as Labour leader at Prime Minister's Questions.