Greenpeace activists said they scaled the Transocean rig Paul B Loyd Junior, contracted to BP, on Sunday evening.
The two Greenpeace protesters have now "set up camp" on the rig, the group said in a statement Monday.
Environmental campaigners have boarded an oil rig in protest at plans to drill for 30 million barrels of oil in the North Sea. BP says they care about climate change, yet they're pumping billions into the search for new oil while lobbying against tougher action on climate change.
If BP does not do as it demands, Greenpeace should "wind down its operations, return cash to investors and go out of business".
TWO Greenpeace activists today vowed to carry on blocking a 27,000-tonne oil rig in the Cromarty Firth operated by BP after spending a sleepless night perched on a gantry on one of the platform's legs.
It says scientists have been clear that we already have more oil and gas than we can safely burn under the Paris Agreement if we want to limit catastrophic climate change "yet BP maintains its desire to both explore for more and expand its oil and gas production".
The protesters are calling for BP to end drilling for new oil wells and say they are prepared to stay on board the rig "for days".
'That means the government must seriously reform the Oil and Gas Authority and instead invest heavily in the crucial work of helping oil communities like those in Scotland move from fossil fuels to the industries that will power our low carbon future, ' she added.
BP said it was working with the rig owner Transocean and the authorities to try to resolve the situation.
But it warned: "While we recognise the right for peaceful protest, the actions of this group are irresponsible and may put themselves and others unnecessarily at risk".
A spokeswoman for BP claimed the company was playing its part and had cut its emissions by 1.7 million tonnes previous year.
A Police Scotland spokesperson told MailOnline: 'In relation to the ongoing protest involving Greenpeace in the Cromarty Firth, Police Scotland is working with the operators, the port authority and other interested parties in an effort to resolve the situation as safely as possible'. We support the Paris agreement. "We are committed to being part of the solution to the climate challenge facing all of us".