The Crown Prosecution Service (CPS) said on Friday that the letter, titled "Cox was first you are next", sent by the now jailed Alden Bryce Barlow arrived at Soubry's constituency office in Broxtowe, Nottingham, on October 14 and was opened by a member of staff, The Daily Mail reported.
Barlow referred to Soubry as "treacherous" and "worthless".
On Friday, he admitted to sending the letter conveying a threatening message and was sentenced to 12 months in jail at the Sheffield Crown Court.
He was also given a 10-year restraining order against Soubry, who is standing as a candidate for The Independent Group for Change (TIGfC) in the December 12 general election.
Barlow was traced using fingerprint analysis and CCTV from the post-office counter in Doncaster from where he had sent the letter.
Chief Crown prosecutor Gerry Wareham called the letter "sickening" and "ominous" and described it as an "attack on democracy".
"Soubry and her staff in the constituency office understandably found the message deeply disturbing and highly offensive.
"I hope the sentence Barlow received today is of some comfort to Soubry and her colleagues and a deterrent to anyone else contemplating such despicable actions against a parliamentary representative or candidate," The Daily Mail quoted Wareham as saying on Friday
Cox, who was Labour a MP for Batley and Spen, was stabbed and shot in Birstall, West Yorkshire, in 2016.
( With inputs from IANS )