The BBC reports the unidentified Trump supporter twice nearly knocked Skeans and his camera over as he "shoved and swore" at other members of the press and yelled "f**k the media" before security escorted him away.
Mr Trump's "Finish the Wall" rally in El Paso was the first dueling rallies of the 2020 campaign season.
As the camera appears to visibly bounce around for about 10 seconds, the screen then focuses on a man in a red cap who is being restrained by another man in a black t-shirt. He added he had previously been "spat at" during a Trump rally.
BBC's Ron Skeans was reportedly attacked and shoved to the ground by an unidentified supporter and Time photojournalist Chris Morris was grabbed by the neck and shoved to the ground by a Secret Service agent during a separate incident.
It comes amid a series of stinging attacks on the press by Trump, branding the "fake news" the "enemy of the people". Fortunately our cameraman is fine, he is made of stern stuff, ' O'Donoghue told the Today programme.
During Monday night's rally in El Paso, the President encouraged support for his proposed border wall with Mexico to protect the United States against a "tremendous onslaught" of migrant caravans, violent criminals and drugs.
O'Donoghue said the goading of the crowds against the media is a "constant feature of these rallies".
Trump has been openly hostile to the press throughout his presidency - accusing them of publishing false stories on him.
"The crowd had been whipped up into a frenzy against the media by Trump and other speakers all night".
Speaking to a packed stadium that holds about 8,000 in El Paso, Trump slammed the 'dishonest media'.
Democrats are scrambling to find a candidate to unseat the Republican - with Trump saying he plans on seeing out his presidency until 2025.
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