Lawyers acting for Richard Tice are threatening to sue Mr Smith for claiming the Brexit Party was "a shell company" that was a "money laundering front" as he gave a TV interview after being re-elected in the European elections.
In a pre-action letter sent to the MEP, legal firm Wedlake Bell said the comments represent "a direct assault on the integrity of our client and strikes at the heart of his professional and business reputation".
Although Mr Smith did not mention Mr Tice by name, the law firm claims the Brexit Party chief "has become a very prominent figure with a high public profile", and added: "There is clearly a strong reference innuendo identifying Mr Tice as the subject of the accusations."
The pre-action protocol letter to Mr Smith effectively acts as a final warning before legal proceedings are started.
Mr Tice has demanded that, to prevent legal action, Mr Smith pays "substantial damages" to compensate for damage to his reputation, covers legal costs, and makes a statement in the High Court in London unreservedly withdrawing the comments and pledging not to repeat the allegations.
The law firm has given Mr Smith a deadline of 5.30pm on Tuesday June 11 before High Court proceedings are started.
Mr Smith referred to the case of Goldsmith and Another v Bhoyrul and Others in 1997, which established the legal precedent that political parties do not have the power to sue in defamation proceedings.
He also called for a "full, open, and transparent independent inquiry" into the funding of Nigel Farage and the Brexit Party.
The remarks were broadcast on Sky News in the early hours of Monday May 27, before the official results from the European election in Scotland were announced, in which the Brexit Party won the second-highest share of the vote (14.8%) behind the SNP which had 37.8%.
The SNP had three MEPs elected, including Mr Smith, compared with the Brexit Party's one European representative for Scotland.
Mr Tice, who founded Leave Means Leave, was elected for the first time as an MEP in the 2019 election to represent the East of England constituency.