Police said that the call to the BBC switchboard was anonymous. However it is believed from sources that the threat was directed against Tony Hall, the head of BBC News.
Police are taking the threat seriously, although there is no way of knowing if they were made by Ms Dando's killer or anyone linked with him.
Two or three calls have been received by the BBC switchboard since Ms Dando's death, mentioning Tony Hall by name.
Mr Hall, a contender for the post of BBC director-general, does not have a high public profile but was one of the first to pay tribute to Ms Dando after her murder on Monday.
A BBC spokesman said last night: "It is not unusual for there to be threats to presenters or people in the BBC.
"We obviously take all of them seriously, but we are not prepared to discuss any detail or any security arrangements."
Mr Hall was supposed to have attended last night's Sony Radio awards, but was advised to stay away by police.
The revelation of the threat to Mr Hall will fuel the theory that a Serbian faction seeking revenge for Nato's bombing of Serb media could be behind Ms Dando's murder.
She led a televised appeal for Kosovan refugees recently.
Mr Hall is ultimately responsible for all journalists working in radio and television news for the domestic BBC.
A Scotland Yard spokesman said: "We can confirm that a threatening call was received by the BBC switchboard yesterday morning by an anonymous male. We are not prepared to discuss it further. It is a line of inquiry being investigated by the Jill Dando investigation team."
Detectives are also investigating whether Ms Dando's killer had tricked his way into her home weeks earlier. It emerged yesterday that Ms Dando's house in Fulham, west London, had recently been sold and that dozens of people had probably been shown around the property in recent months. Police are to examine the estate agent's files and are expected to question some of those who viewed the house.
Meanwhile, police are preparing to release an e-fit of the suspect seen near Ms Dando's Fulham home on the day of the shooting.
Last night's edition of the television programme Crimewatch File, to which Ms Dando contributed, was postponed.Reuse content