The two children, understood to be siblings – Kiwi, 12, and Zozo, ten – abseiled down the structure around 2:50pm on Monday afternoon.
Multiple police vans and the coastguard were seen at the scene.
It is understood the children’s parents were arrested and are currently in custody.
One man aged 45 and a woman aged 43 were arrested in connection with an alleged offence under the Children and Young Persons Act following what police described as a “peaceful protest” near to the Clyde Arc.
A spokesman for the children said they were safe and well and having dinner and playing cards with a friend.
They are staying in Glasgow.
The children, named as Kletter Kinder (Climbing Kids) on Twitter, were hanging from the site for roughly half an hour as a symbolic message to raise awareness for the climate change crisis.
Dangling from the well-known Glasgow bridge, the children held up a sign saying ‘Humanity is Failing’ in what they have described as “a call to action against global warming and for the 1.5C goal at COP26”.
One of the children, ten-year-old Zozo, said: “Humanity is failing. We are failing to accept the reality of the climate crisis.
"We're racing towards a wall and our politicians are celebrating that they want to brake behind the wall.”
Her brother Kiwi added: “The IPCC reports are brutal. The global CO2 budget for 1.5C will be used up in a few years. After 1.5C, a cascade of tipping points will be triggered.
"Millions of people will die, billions will loose their home or won’t be able to feed their families.”
The two children have done climbing for many years and have done similar protests in Germany and also protested at COP25.
A Police Scotland spokesperson said: "Officers attended a peaceful protest near to the Clyde Arc this afternoon.
"A man aged 45 and woman aged 43 were arrested in connection with an alleged offence under the Children and Young Persons Act."