News of the mystery buyer came as islanders fought to stave off an attempt by a previous owner to remove a 200-year-old map from the island.
It is the second time they have tried to buy the Scottish island which has been up for sale since last summer. Their first bid in November was rejected but they have since "significantly increased" their first offer of pounds 1.4m.
The island, which has only 63 inhabitants, was put on the market last summer by the German artist Marlin Eckhard Maruma, for pounds 2m. He bought it in 1995 from Keith Schellenberg, a Yorkshire businessman.
Despite the fact that Mr Schellenberg left the island more than two years ago residents are still fighting to stop him removing a priceless map of the island which has been there for 200 years. He tried to take the map with him when he left, but furious islanders barricaded the craft shop where the map was awaiting transportation to the mainland, and "incandescent with rage", he was forced to leave without it.
Undeterred, he took out a court action to establish ownership of the map. Mr Maruma claims there was a gentleman's agreement between the two men that the map and other documents belong to the owner and should not be removed from the island. But when the case came to court last month Mr Maruma did not turn up and Mr Schellenberg was able to lay legal claim to it.
The map is now in the hands of Eigg's special police constable, Colin Carr. Chief Inspector John McFadzean, of Fort William police, said the court had ordered the map to be handed over to Mr Schellenberg as the legal owner. "We are trying to comply with that order but we are having certain logistical problems," he said.
Maggie Fyffe, secretary of the Isle of Eigg Heritage Trust, which is trying to buy the island in partnership with the Highland Council and Scottish Wildlife Trust, said the map belonged on the island.
"It would be blatant asset-stripping of the worst kind if something as important as the map, which is part of Eigg's heritage leaves the island," she said. "The island has changed hands many times in the two centuries which have intervened but not one owner has tried to take it with them so what gives Keith Schellenberg the right?
But even if they manage to fight off Mr Schellenberg the islanders are worried about the island passing into the hands of another wealthy but absent landowner. The mystery buyer is rumoured to be a semi-retired farmer and property dealer, Graham Mellstrom, from Hampshire. Mr Mellstrom is currently in Germany and could not be contacted.
His son, Stephen, said he could not confirm whether his father had put in a bid. "I know nothing about his plans," he said. "He keeps that side of his life very much to himself ..."Reuse content