It means that England captain Farrell, who has led England at the last two World Cups and won 112 caps, will miss this season's Six Nations Championship.
In his absence - no date has been announced for a Test return - and following the international retirement of Courtney Lawes, Bristol prop Ellis Genge is a probable successor as England skipper.
George Ford would be favourite to take Farrell's No.10 role for the Six Nations opener against Italy in Rome on February 3, with Marcus Smith also a contender.
Borthwick said: "Everyone at England Rugby is fully behind Owen's decision.
"Since making his debut, he has been an integral part of the England set-up for over a decade, and the demands on elite athletes are extremely challenging.
"He is an exemplary player, captain and leader and always gives his all for his country.
"It is with typical courage that Owen has made this decision to open up in this manner.
Everyone at England Rugby stands behind Owen's decision to prioritise his and his family's mental well-being — England Rugby (@EnglandRugby) pic.twitter.com/0zrIqCn38ANovember 29, 2023
"Together with all of us at England Rugby, I will do everything I can to ensure that he has the support he requires going forward."
In a statement announcing the surprise - and sobering - development, Farrell's club Saracens said: "Owen Farrell has decided to take a break from international rugby in order to prioritise his and his family's mental well-being.
"This means he will not be available for selection for the 2024 Six Nations.
"He will continue to play for Saracens and captain the club.
"As always, Owen will have the full support of everyone at the club."
The Rugby Players' Association, meanwhile, added: "He is a leader and figurehead in the English game, but is also an individual, husband and father. His well-being comes first, and we will support him in every way we can, going forward."
Farrell led England to a third-place World Cup finish in France last month, but the tournament build-up proved far from plain sailing for him.
He was sent off in a World Cup warm-up game against Wales, only for an independent disciplinary panel to cause an outcry when it cleared him following a shoulder-led tackle to the head of Wales forward Taine Basham, which was expected to result in a significant ban.
World Rugby then appealed that decision, and he received a four-match ban that included England's opening two World Cup fixtures against Argentina and Japan.
Farrell's father - Ireland head coach Andy Farrell - labelled media coverage of the episode "a circus", while Borthwick said in August: "The commentary around it seems to move from an issue around the tackle to personal attacks on the character of the man, which I think is just wrong."
Farrell, 32, was subjected to considerable attacks on social media, and at times during the World Cup he was booed by sections of the crowd during England games.
Having made his England debut in 2012, Farrell has amassed an England record 1237 points, recently overtaking Jonny Wilkinson.
He captained England to the final at the World Cup in 2019 and to the semis in 2023 and has also featured on three British and Irish Lions tours.
