In a statement, the corporation says one reporter has been "attacked on social media by the mayor of Ankara for her coverage of the current protests".
On Sunday, Mayor Ibrahim Melih Gokcek described BBC Turkish reporter Selin Girit as an "English agent", launching a campaign against her on Twitter.
This triggered a counter-campaign which became Turkey's most trending topic.
In the statement on Monday, BBC Global News Director Peter Horrocks said that "a large number of threatening messages have been sent to one of our reporters".
He stressed that BBC journalists "are committed to providing impartial and independent journalism" and must not "be directly targeted in this way".
The BBC and Turkey have previously clashed over the corporation's coverage of the mass protests
"There are established procedures for making comments and complaints about BBC output and we call on the Turkish authorities to use these proper channels," the statement added.
In a separate statement, Britain's National Union of Journalists (NUJ) said: "We want to send a strong message to Turkish authorities - it is simply not acceptable to target journalists in your turbulent times. We condemn the attempts to intimidate journalists and the threats must stop immediately."
Responding to the BBC accusations, Mr Gokcek said it was "unacceptable that Turkey is targeted by the BBC".
The mayor, a member of the ruling AK party of Prime Minister Recep Tayyip Erdogan, began the Twitter campaign against Ms Girit on Sunday, accusing her of trying to undermine the Turkish economy in her reporting.
He urged his followers to denounce the BBC journalist by sending tweets to the newly-created hashtag, which could be roughly translated as "Don't be an agent on behalf of England Selin Girit".
This soon became one of the most popular Twitter topics in Turkey.
But it also angered many people across the country. They countered by using the hashtag "Melih Gokcek is a provocateur ", which quickly rose to become the country's number one trending theme.
The mayor is now threatening to sue every user tweeting with the hashtag.
This is not the first time that the BBC and Turkey have clashed over the corporation's coverage of the continuing anti-government protests in Turkish cities.
Earlier this month, the corporation said it was suspending its partnership with Turkey's NTV television channel, following NTV's decision not to transmit the BBC current affairs programme Dunya Gundemi [World Agenda]".
The BBC said that "any interference in BBC broadcasting is totally unacceptable and at a time of considerable international concern about the situation in Turkey the BBC's impartial service to audiences is vital". (bbcnews)