In a statement accusing Mr Bourke of "blatant and mischievous lies" by suggesting they had threatened him with violence, the politicians hinted a compromise had been reached whereby he would not remain beyond the minimum three-year term. He took over in 1993 with an open-ended term.
The statement, signed by the Chief Minister, Derek Taylor, and the opposition leader, Washington Misick, said: "We believe it to be in the interests of all parties, particularly Mr Bourke, that he should be allowed to leave at the earliest possible opportunity." It was implied local leaders would refuse to co-operate with him during the rest of his term.
Commenting on a Foreign Office statement that they had threatened violence, the politicians said: "Violence will not form part of this struggle. We may not be on hand to wish him farewell but we are determined that Mr Bourke should leave in good health."
Mr Bourke, 49, angered most of the 15,000 islanders earlier this year when he spoke in an interview of drug-trafficking, rising crime and corruption. The politicians said his comments would hinder badly needed investment, particularly in tourism and offshore finance. Mr Bourke also upset many when he was flown on a private jet to Miami last month for an appendicectomy rather than be operated on here.
At last week's Whitehall meeting the Foreign Office rejected demands for his recall.
Questioned about yesterday's statement, locals said they thought a compromise under which the Governor would leave at the end of the minimum term was the only face-saving solution for both sides.
In an interview at the weekend with the Independent, Mr Bourke said he was concerned for his safety and that of his wife, Anne Marie. "Violence is not unknown here," he said. "Burning down buildings is not unknown. When you have drugs and politics, nasty things happen." Mr Bourke used to be seen regularly around Grand Turk, the island which is the seat of government, in his "limousine" - a white-painted London taxi."You'd see him from time to time in the Turk's Head bar or the Nookie Hill Club," said one resident. "But he's not venturing out much any more."
Mr Bourke, whose main duty is to take part in the weekly cabinet meeting, spent Easter trying to arrange the repatriation of 70 people from the Dominican Republic who arrived illegally last week. Growing numbers of people from the Dominican Republic and, more so, Haiti, have become the Turks and Caicos's main problem.
Officials say there may be about 6,000 illegal Haitians here, along with 5,000 legals, making the Haitian contingent almost as big as the population of natives and "belongers" (expatriates granted citizenship).
Locals blame the Haitians for much of the rise in crime, mostly burglaries. Officials say there is a Haitian mafia in the Haitian port of Cap-Haitien which charges people about $700 (pounds 460) for the trip, dropping them off in the shallow waters off the coast of the island of Providenciales and often telling them they are in Florida.