The Shard, so named for its broken-glass design, is 1,016-feet tall landmark with 87 storeys.
The protesters are believed to have accessed the building in central London by climbing on to the roof of nearby London Bridge Station in the early hours today.
"We were called at 4.20 am (local time) today to a group of protesters attempting to climb up the Shard. We are in attendance and monitoring the situation along with the British Transport Police," Metropolitan police said.
The campaigning group said the climbers would stay on the building as long as it was safe to do so, and that they are prepared to be arrested.
More From This Section
A spokesman said all six were well trained and if they manage to get to the top of the building, they plan on installing a piece of art.
"Our primary focus is on the safety of the protesters and the workers and visitors to the building.
The Qatari-owned building, which was opened this February, contains offices, restaurants, a hotel and residential apartments, and the capital's highest viewing gallery, The View from The Shard, offering 360 degree views.
Greenpeace, which is known for headline-grabbing stunts around the world, is campaigning for the area around the North Pole to be made a global sanctuary, off limits to industrialisation.
"We respect the right of individuals and organisations to engage in a free and frank exchange of views about our operations. We only ask that they do so with their safety and the safety of others in mind," said Shell.