Andrew Mitchell's lawyers sent The Sun a "letter before action" last month, putting it on notice that the Conservative MP intended to sue.
At the time, his lawyer Graham Atkins told The Guardian: "We are seeking damages, an apology, an undertaking that the words complained of are not repeated, and costs."
The newspaper was the first to break the story of Mr Mitchell's row with the officers at Downing Street, which became known as the "Plebgate" scandal.
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The operation to recover the unexploded bomb near the Houses of Parliament let to disruption in central London overnight.
Southern parts of England and Wales will see sunny skies after a frosty start.
A police spokesman said the incident in the Bourke Street Mall on Friday was not related to terrorism.