The Stena Carron is currently at anchor in Bressay Sound as Chevron waits on permits to drill a wildcat well on the Lagavulin prospect, about 260 kilometres north of Shetland.
Yesterday, two activists climbed one of the drillship's anchor chains and set up camp on a small platform suspended about five metres above the sea.
Two other activists, who were both involved in the environmental group's action targeting Cairn Energy's operations off Greenland earlier this month, are in the pod, and say they can continue their occupation for a month.
Chevron was not immediately available for comment. However yesterday the US supermajor called on Greenpeace to call off the action, branding it "foolhardy" and saying it as "concerned for the safety of those involved".
The supermajor added it was willing to meet with Greenpeace to address their concerns and explain its preparations to ensure its drilling operations are safe and environmentally responsible.
Greenpeace said the action is part of its campaign to halt deep-water drilling in environmentally sensitive areas.
The Northern Constabulary is currently monitoring the situation. A spokesman said the force was working with partner agencies to resolve the incident peacefully.
Yesterday Chevron said the protest would not affect its drilling schedule, as it is still waiting on permits for the Lagavulin well and will not be mobilising until these are secured.
The supermajor is targeting more than 500 million barrels of oil in Block 217/15, classing Lagavulin as one of the largest undrilled structural closures on the Atlantic Margin.
Drilling is expected to last about 109 days and success could lead to drilling on the higher-risk Talisker prospect in blocks 217/a and 4a.
Chevron holds a 60% operating stake in the licence, where it is partnered by OMV on 20%, and Faroe Petroleum and Idemitsu E&P each with 10%.
During the Cairn action, four Greenpeace activists occupied the semi-submersible drilling rig Stena Don during operations on the Sigguk Block. The protest stopped drilling for two days.
Greenpeace is also threatening legal action against the Cameron government in an effort to stop the granting of new permits for deep-water drilling off the UK.
Last month Greenpeace lawyers sent a "letter before action" to ministers – the precursor to seeking a judicial review of the decision to push ahead with new deep-water drilling off the UK.