The Tesla founder is said to have made the offer in May 2016, during a text message exchange in which Ms Heard, 34, told him she was going to obtain a restraining order against Johnny Depp.
In her second day of evidence in her ex-husband’s blockbuster libel action against The Sun newspaper, the Aquaman actress also dismissed suggestions Mr Musk visited her in 2015 while Mr Depp was away, saying she was not in touch with the entrepreneur until the following year.
Mr Depp, 57, is suing the tabloid’s publisher, News Group Newspapers (NGN) and its executive editor, Dan Wootton, over a 2018 article which labelled him a “wife beater”.
On Tuesday, Eleanor Laws QC, representing Mr Depp, read out text messages between Ms Heard and Mr Musk from May 22 2016, in which Ms Heard tells Mr Musk that she is going to obtain a restraining order against her then husband.
This is the day after an alleged incident at the Eastern Columbia Building in Los Angeles, where Mr Depp and Ms Heard lived, in which the actress claims Mr Depp threw her mobile phone at her, hitting her in the face – which the actor denies.
Questioning Ms Heard, Ms Laws said Mr Musk offers to “arrange 24/7 security for you”, adding: “The offer would stand, even if you never wanted to see me again… anyway, sorry for being an idiot. The radio silence hurts a lot. It only matters because I really like you.”
The barrister then referred to evidence given by Alejandro Romero, a concierge at the Eastern Columbia Building, who said he saw Mr Musk visiting Ms Heard “when Mr Depp was in Australia”.
Ms Heard responded: “He’s wrong. I wasn’t even in communication with Elon until 2016.”
In his written witness statement, Mr Romero said: “From March 2015 onwards, Ms Heard was visited regularly late at night, at around 11pm to midnight, by Mr Elon Musk.”
The concierge said Ms Heard gave Mr Musk his own fob to access the building.
Ms Laws also asked Ms Heard about actor James Franco, saying: “On the topic of what you say about his (Johnny Depp’s) jealousy, you mentioned in your statement that James Franco was someone that Mr Depp was accusing you of having an affair with and he was quite wrong about it.”
Ms Heard said: “That’s right.”
Ms Laws then played CCTV footage to the court of Ms Heard travelling downstairs in the lift at the Eastern Columbia Building on May 22 2016 at around 11pm, before travelling back upstairs shortly after with Mr Franco.
Ms Heard confirmed that the man in the footage was Mr Franco.
Ms Laws asked: “Are you trying to avoid being seen by the camera there?”
Ms Heard replied: “We were talking. He was saying to me ‘oh my god, what happened to you?’
“He saw my face… when I let him in. He saw my face and he said ‘what the f***?'”
Ms Laws asked: “That’s not secretive talking on that camera?”
Ms Heard said: “Yes, it’s discreet.”
Asked about what time Ms Heard collected Mr Franco from downstairs, Ms Heard confirmed it was after 11pm and said: “In those days, I didn’t sleep much at night.”
Ms Heard is continuing to be questioned over 14 allegations of domestic violence, which the tabloid’s publisher News Group Newspapers (NGN) relies on in its defence against Mr Depp’s claim.
She has accused Mr Depp of verbal and physical abuse throughout their relationship, including screaming, swearing, punching, slapping, kicking, headbutting and choking her, as well as “extremely controlling and intimidating behaviour”.
Mr Depp, however, says he was “not violent towards Ms Heard – it was she who was violent to him”, and claims his ex-wife was “building a dossier” of false allegations as an “insurance policy for later”.
Ms Heard’s evidence was initially due to conclude on Wednesday but will now continue until Thursday morning, with her friends Melanie Inglessis and Joshua Drew expected to appear by videolink on Wednesday afternoon.
Mr Depp is suing NGN and The Sun’s executive editor Dan Wootton over the publication of an article on April 27, 2018, with the headline: “Gone Potty: How can JK Rowling be ‘genuinely happy’ casting wife beater Johnny Depp in the new Fantastic Beasts film?”
His lawyers say the article bore the meaning there was “overwhelming evidence” Mr Depp assaulted Ms Heard on a number of occasions and left her “in fear for her life”.
NGN is defending the article as true, and says Mr Depp was “controlling and verbally and physically abusive towards Ms Heard, particularly when he was under the influence of alcohol and/or drugs”.