Scotland Yard confirmed it was looking into allegations made on Wednesday, when Hall confirmed he would not be renewing Clarkson's contract because of his unprovoked attack on producer Oisin Tymon at a North Yorkshire hotel.
The Mail on Sunday reported that the director-general - officially crossbench peer Lord Hall of Birkenhead - and wife Cynthia had been under 24-hour guard since the threat was received.
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A Scotland Yard spokesman said: "Police in Westminster are investigating an allegation of threats to kill. The allegation was reported to police on Wednesday March 25. Inquiries continue. No arrests have been made."
The Mail on Sunday printed pictures showing security guards on patrol outside Hall's Oxfordshire home, claiming it was costing £1,000-per-day to protect the BBC's top executive. It also said that the BBC's head of security had asked for ex-special forces soldiers to join the team.
Hall, former chief executive of the Royal Opera House, took over the £450,000 BBC post in April 2013 to replace George Entwistle, who left the corporation in the wake of the Jimmy Savile sex abuse scandal.
On Wednesday Hall announced that Clarkson, a popular but divisive figure during his time at the hugely successful BBC 2 motoring show, would not be retained, saying ''a line has been crossed'' and that ''there cannot be one rule for one and one rule for another''.
Tymon had his lip split by Clarkson in a 30-second-long assault on March 4 and took himself to hospital with his injuries. He was also shouted at by the former Top Gear presenter in a torrent of verbal abuse.
Clarkson reported the incident to the BBC five days later and was suspended by the broadcaster on March 10.
A BBC spokeswoman said: "We wouldn't comment on security matters."
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