The cease-and-desist notice accuses Mr Bannon of defaming the president in speaking to author Michael Wolff.
Mr Wolff's forthcoming tell-all book describes the president as being unprepared for the job.
Mr Trump responded by saying Mr Bannon had "lost his mind" after losing his White House position.
His lawyers said Mr Bannon had broken his employment agreement by speaking to Wolff about Mr Trump and his family, "disclosing confidential information" and "making disparaging statements and in some cases outright defamatory statements to Mr Wolff about Mr Trump, his family members, and the Company", the letter said.
Trump-Bannon: A knife fight with consequences.
What is in the book?
The book - Fire and Fury: Inside the Trump White House - makes many explosive claims, including that:
Mr Bannon thought a meeting between Donald Trump Jr and a group of Russians was "treasonous"
The Trump team was shocked and horrified by his election win
His wife, Melania, was in tears on election night
Mr Trump was angry that A-list stars had snubbed his inauguration
The new president "found the White House to be vexing and even a little scary".
His daughter, Ivanka, had a plan with her husband, Jared Kushner, that she would be "the first woman president"
Ivanka Trump mocked her dad's "comb-over" hairstyle and "often described the mechanics behind it to friends"
10 explosive claims from new Trump book
It also alleges that former UK Prime Minister Tony Blair told Mr Trump during a meeting last February that the British intelligence services may have been spying on him and his campaign, according to a report in The Times newspaper.
The Times says Mr Blair was hoping to get a job advising Mr Trump on the Middle East. A spokeswoman for Mr Blair told The Times that the allegations were a "total fabrication" and said Mr Blair had not suggested himself as Mr Trump's Middle East envoy.
In March 2017, then-White House press secretary Sean Spicer told journalists UK intelligence services could have been involved in an alleged spying operation on Trump Tower in New York. GCHQ said the allegation was "nonsense"..