The attack came just days after repair work began on the previously-attacked Grade II listed statue.
The statue of Prospero and Ariel from Shakespeare’s The Tempest was created by Eric Gill, who is said to have sexually abused two of his daughters.
The man, wearing a Spiderman mask, was seen on scaffolding surrounding the sculpture on Saturday where he also strung up a banner criticising the corporation.
The statue has been on display at Broadcasting House since 1932 but was attacked last year amid calls for it to be removed.
The Metropolitan Police were called at around 4.15am on Saturday to reports of a man climbing scaffolding at premises in Portland Place.
Speaking of the protest, a Met Police spokesperson said: “The man was reportedly damaging a sculpture.
“Officers attended the location.
“It has not been possible to safely detain the man given the circumstances of the incident, including the height.
“Specialist officers have been called to the scene.”
On Tuesday, the BBC said after “careful consideration” taking into account the historical and cultural significance of the building and after discussions with leading cultural organisations such as Historic England, the repair work was agreed with all costs covered through the corporation’s insurance and not the licence fee.
The scaffolding went up around the building on Tuesday with expert stonemasons beginning to restore the work, carved from Caen stone, a type of limestone quarried in north-west France.
The broadcaster said the repair work would be used to provide additional context about the artwork and Gill, and that members of the public would also be able to access a QR code nearby.
The head of BBC History, Robert Seatter said: “Broadcasting House is a building of historical and cultural significance and one of the foundations of modern-day broadcasting, both in this country and around the world.
“We have a responsibility to maintain and preserve the building for generations to come.
“Alongside this, Gill’s abusive behaviour and lifestyle are well documented and the BBC in no way condones his behaviour.
“So while it is right that the fabric of the building is restored, we must also ensure people are fully informed about the history connected to it.”
More than 3,000 people have signed a petition demanding the removal of the sculpture on the website of the political activist group 38 Degrees.
The repairs had been set to be completed by the end of June.