The Forties pipeline system, which is one of the oldest and largest in the North Sea, is about 100 miles long and transports around 450,000 barrels of oil a day, which is about 40 per cent of UK production.
Spokespeople from both companies confirmed the discussions.
"At the moment the details of these conversations are confidential and we cannot say any more at this stage," said a spokesperson from Ineos.
BP said: "We remain committed to communicating openly with staff and our stakeholders as soon as we are able, and as commercial confidentialities allow, if any deal is confirmed or agreed."
The oil major has been searching for a buyer for more than a year as it looks to sell older North Sea assets.