The six women climbing up The Shard are from UK, Canada, Sweden, Poland, Holland and Belgium.
Their daring ascent started from the roof of London Bridge at 4am on Thursday.
Greenpeace said two lead climbers were 'free climbing' - scaling the building without help - and they were using their fingertips to pull themselves up, fixing safety ropes as they go to help the rest of the group.
The group are carrying a "huge work of art" in backbacks, Greenpeace said.
One woman, 32-year-old Canadian Victoria Henry, lives in Hackney.
Before beginning the ascent, she said: "It’s going to be really hard work, it’s going to be nerve-shredding for all of us and we may not succeed, but we’re going to do everything we can to pull it off."
Another woman, Sandra Lamborn, 29, from Sweden, said: "It’s crazy. I’ve come to London to make a stand. We’re drawing a line in the ice and saying to the oil companies, you come no further.”
One climber, Victoria Henry, wrote a blog in advance of the stunt.
She said she had been climbing for years, but this was her "biggest ever challenge".
Henry, who said she had trained in "non-violent direct action", said: "With any luck, as you read this I’ll be clinging to the side of the Shard, hundreds of metres up in the sky.
"But as I write this, with less than a week to go, I’m just feeling... tired.
"I have sores on my shoulders from training with backpacks full of weights, and every night brings tiresome dreams about carabiners and tangled ropes.
"But I wouldn’t have it any other way. I’m training for something that I hope will help stop one of the most heartbreaking acts of wanton environmental destruction - drilling for oil in the Arctic.
"I’m over the moon that I can play any part in the mass resistance to this horrifying practice."