Northumbria Police said the boy has since been released on police bail, pending further inquiries.
The tree, also known as the “Robin Hood Tree”, featured in the 1991 Hollywood film Robin Hood: Prince Of Thieves, which starred Kevin Costner and Morgan Freeman.
Members of the public have been asked to avoid the site, near Crag Lough, while work is under way to make the area safe.
While the tree was reported to be 300 years old, the National Trust said it was planted between 1860 and 1890.
One expert told Sky News it was unlikely the tree – voted English Tree of the Year in 2016 in the annual Woodland Trust awards – could be saved.
John Parker, chief executive officer at The Arboricultural Association, told Sky News: “There is a chance you might get shoots at the bottom – but the tree will never be able to re-establish itself to the way it was before.”
However John Stokes, from The Tree Council, said it was “worth having hope”.
He told Sky News: “At this time of year, trees begin to store energy in their roots for next year’s growing season – and it is possible that the tree may grow some new shoots next spring.”
Northumbria Police and Crime Commissioner Kim McGuinness said she was “devastated that the famous Sycamore is gone”, describing it as “an iconic North East landmark”.