Scotland Yard confirmed it was looking into allegations made on Wednesday, when Mr 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 director-general – officially crossbench peer Lord Hall of Birkenhead – and wife Cynthia have reportedly been under 24-hour guard since the threat was received.
A Scotland Yard spokesman said: “Police in Westminster are investigating an allegation of threats to kill. The allegation was reported to police on Wednesday 25 March.
“Inquiries continue. No arrests have been made.”
Security guards have been seen on patrol outside Mr Hall’s Oxfordshire home, amid claims it is costing £1,000 per day to protect the BBC’s top executive. The BBC’s head of security is thought to have asked for ex-special forces soldiers to join the team.
Mr 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, Mr Hall announced that Clarkson, a popular but divisive figure during his time at the hugely successful BBC2 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”.
He said the BBC would look to “renew” the show for next year and look at ways to screen the remaining episodes of the current series, cut short following the fracas that cost the 54-year-old his job.
Mr Tymon had his lip split by Clarkson in a 30-second-long assault on 4 March 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 10 March.
A BBC spokeswoman said: “We wouldn’t comment on security matters.”
Meanwhile, BBC Radio 2 DJ Chris Evans said axeing Clarkson as host of Top Gear is not a “fatal blow” for the show.
Evans, who remains among bookmakers’ favourites to take Clarkson’s place despite repeatedly ruling himself out of the running, hit back at claims the hit BBC2 motoring show would need to be “reinvented”.
He said rumours co-hosts James May and Richard Hammond were also considering their positions were “wrong”, backing the pair to keep the series going.
Evans, writing in a Sunday newspaper, said: “In no way does Top Gear need to be reinvented. This series was shaping up to be the best yet. After surviving the insanity of what happened in Argentina for the Christmas special, it felt like the whole team had been drawn closer together as a result.
“As for Jeremy no longer being on the show, sure it’s a huge loss. Top Gear was the show he was born to do. But by no means is it a fatal blow.”
Politics, music and football all survived the departures of Winston Churchill, John Lennon and George Best, Evans said.
Reiterating he would not be taking part in the show, the One Show presenter said: “I love Top Gear. But I never want to watch me on it.”
He added: “I also happen to think I actually wouldn’t be very good at it.”
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