The BBC is facing calls to investigate claims Dominic Raab swore at young staff member.
In an article in The Independent, prominent anti-Brexit activist Gina Miller said she saw Mr Raab tell the man to “go get me a f***ing car”.
The alleged incident took place after Ms Miller and Mr Raab appeared together on the Radio 4 Today show in 2016.
The BBC is now under pressure to investigate the claims and “give support” to the young staffer.
A source close to Mr Raab has described the claims as “baseless and malicious”.
Ms Miller claimed in her article that the deputy prime inister launched an “abusive attack” on her.
She said she was “bullied and demeaned” by Mr Raab after he called her “stupid” and “naive” during an “aggressive” encounter at the BBC.
Her claims are the first on-the-record accusations of abuse against Mr Raab, who faces an official inquiry into allegations that he bullied civil servants.
Ms Miller wrote: “He was aggressive and intimidating, and I was bullied and demeaned. This was an aggressive male expressing seemingly misogynistic behaviour. This sort of behaviour is not acceptable from anyone, especially not from a powerful, influential politician.”
Referring to the alleged incident involving the BBC runner, she claims that he “barked” at the employee “Where’s my car?” When the young man replied that they had not been asked to arrange one for him, a “furious” Mr Raab shouted: “Go get me a f***ing car.”
Daisy Cooper, Deputy Leader of the Lib Dems, said: “In light of these new allegations, the BBC must investigate the incident and give support to the young staffer.
“If true, the individual should be encouraged to come forward and give evidence to the Tolley Inquiry.
“Liberal Democrats believe that bullying and harassment have no place in our society, especially not at the hands of those entrusted with the rule of law.”
The BBC has been approached for comment.