The BBC is objecting to a new label describing it as “government funded media” on its main Twitter account.
The corporation has contacted the social media giant over the designation on the @BBC account to resolve the issue “as soon as possible”.
“The BBC is, and always has been, independent. We are funded by the British public through the licence fee,” it said.
Elon Musk said he believed the BBC was one of the “least biased” outlets.
When BBC News highlighted to the Twitter boss that the corporation was licence fee-funded, Mr Musk responded in an email, asking: “Is the Twitter label accurate?”
He also appeared to suggest he was considering providing a label that would link to “exact funding sources”.
It is not clear whether this would apply to other media outlets too.
In a separate email seeking to clarify his earlier comments, Mr Musk wrote: “We are aiming for maximum transparency and accuracy. Linking to ownership and source of funds probably makes sense. I do think media organizations should be self-aware and not falsely claim the complete absence of bias.
“All organizations have bias, some obviously much more than others. I should note that I follow BBC News on Twitter, because I think it is among the least biased.”