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."
The level of the £159 ($197) annual licence fee - which is required by law to watch live TV broadcasts or live streaming in the UK - is set by the government, but paid for by individual UK households.
While the @BBC account, which has 2.2m followers, has been given the label, much larger accounts associated with the BBC's news and sport output are not currently being described in the same way.
The account primarily shares updates about BBC-produced TV programmes, radio shows, podcasts and other non-news material.
The label links through to a page on Twitter's help website which says "state-affiliated media accounts" are defined as "outlets where the state exercises control over editorial content through financial resources, direct or indirect political pressures, and/or control over production and distribution".
As the UK's national broadcaster, the BBC operates through a Royal Charter agreed with the government.
The BBC Charter states the corporation "must be independent", particularly over "editorial and creative decisions, the times and manner in which its output and services are supplied, and in the management of its affairs".
Twitter's new labelling of the BBC's account comes after it did the same to US public broadcaster NPR's handle.
Initially the social media firm described NPR as "state-affiliated media" - a label given to outlets including Russia's RT and China's Xinhua News.
The designation was later changed to the same "government funded media" tag now applied to the @BBC account. NPR had said it would stop tweeting from the account unless it was amended.
The licence fee raised £3.8bn ($4.7bn) in 2022 for the BBC, accounting for about 71% of the BBC's total income of £5.3bn - with the rest coming from its commercial and other activities like grants, royalties and rental income.
The BBC also receives more than £90m per year from the government to support the BBC World Service, which predominantly serves non-UK audiences.
The national broadcaster's output is also paid for through the work of commercial subsidiaries like BBC Studios, as well as through advertising on services offered to audiences outside of the UK
By law, each household in the UK has to pay the licence fee (with some exemptions) if they:
TV licence evasion itself is not an imprisonable offence. However, non-payment of a fine, following a criminal conviction, could lead to a risk of imprisonment - "a last resort" after other methods of enforcement have failed.
