In his latest column in the Guardian, Mr Jones claimed The Spectator – which is chaired by Mr Neil – gave “such generous space” to the outgoing US President in wake on the recent impeachment order.
But now, Mr Neil has retaliated claiming Mr Jones is a “compulsive liar” and cannot produce any examples of him “cheerleading for” Trump.
Mr Neil tweeted: “Owen Jones is a compulsive liar.
“He lumps me with so called ‘cheerleaders’ for Trump.
“Yet he cannot produce a single example of me cheerleading for him.
“If he had a reverence for facts he would reproduce my regular criticism of Trump.
“But that would conflict with his distorted narrative.”
Mr Neil continued his attack on Mr Jones who said he holds him “responsible” for what The Spectator writes.
He continued: “He tries to hold me responsible for what The Spectator has written about Trump.
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The former BBC veteran then argues Mr Jones’ claims of “inconsistency” is laughable due to his support of Jeremy Corbyn,
Mr Neil added: “He accuses them of inconsistency.
“For someone who was all over the place when it came to Jeremy Corbyn, that’s hilarious.”
In his column, Mr Jones criticised the Spectator for its coverage of the Trump administration over the last four years.
He wrote: “And yet no British publication gave such generous space to Trump and Trumpism as the Spectator, publishing articles with headlines such as ‘The intelligent case for voting Trump’ and ‘Trump will be much, much better for Britain’, or crowing ‘Donald Trump’s victory marks the death of liberalism’.
“There is a broad consensus that what paved the way for Wednesday’s insurrection in Washington DC was the deliberate (and baseless) delegitimising of the presidential election, and in November, the Spectator was publishing article such as ‘Trump is right not to concede’ and ‘Can you really blame Trump for refusing to accept the election result?’”
Mr Jones’ criticism of the Spectator goes as far back as 2019 when he accused the publication of stroking prejudice.
During an appearance on the BBC’s ‘This Week’ show, Mr Jones found himself the focus of Mr Neil’s ire.
While Mr Jones claimed the magazine had “defended Greek neo-Nazis” and published Islamophobic content, Mr Neil outright refused to allow such comment.
Mr Neil replied: “I knew you were going to bring that up and I won’t let you hijack.
“Your smears about me are not going to be dealt with tonight so just move off.”
The two were discussing extremism in UK politics after Mr Jones and Tory MP Anna Soubry were verbally abused outside Parliament at the time by a group of far-right activists.
The two had previously clashed on Twitter.