Confirmation of the discussions came after weeks of speculation and hours after BP formally announced that Conn, its group managing director and head of Downstream, was to leave the oil group after 10 years on the board.
The top job at Centrica is currently considered one of the toughest jobs in business and Laidlaw, 58, is thought to be leaving after eight years in the job partly because of the antagonism surrounding the company.
British Gas has been a lightning rod for politicians and consumer groups accusing the big six suppliers of profiteering at a time of acute fuel poverty, something Laidlaw has always strenuously denied.
The whole supply sector is now being investigated by the Competition and Markets Authority, following concerns raised by energy regulator Ofgem, and others.
Centrica said in a statement to the stock exchange that "it has been, and continues to be, in discussions with Iain about the possibility of him succeeding Sam Laidlaw as chief executive of Centrica."
But it insisted privately that details on any succession were still not finalised meaning that Conn, 51, had left BP after 27 years without securing a definite job at Britain's largest domestic energy supplier.
Industry sources said the two sides were very close to a deal, however, with high expectations that a formal appointment of Conn will be unveiled by next Thursday at the latest, alongside Centrica's half year financial results.
Conn has a reputation for being articulate, affable but tough. At BP he came in to the refining and chemical division after the Texas City fire and masterminded a huge retrenchment with thousands being taken off the oil company's payroll, including those at Grangemouth in Scotland.
One friend of his said last night. "Ian would be good for Centrica. He has real people skills. He is very good speaker and likes rubbing shoulders with politicians."
Conn will leave after 29 years at BP – including 10 years on the board – with a potential cash pile, partly made up of share options, that could be worth as much as £16m.
Centrica has targeted the tough oil man as an ideal repleacement for Laidlaw but has been wary of attracting bad publicity if it has to buy him out of the massive share option deal. It is believed that Conn has agreed to move to Centrica with some of his BP options being left behind.
BP said it was handing over Conn's role of chief executive, downstream, to Tufan Erginbilgic on 1 October. Erginbilgic is currently chief operating officer for BP's fuels businesses, which includes responsibility for BP's global refining function.
Bob Dudley, BP chief executive, said: "Iain has led the repositioning of BP's Downstream business over the past few years, transforming and greatly strengthening both its portfolio and performance – financially and also in terms of safety and operations. He has been an invaluable member of the executive team and, for me personally, a great support. He leaves with my very best wishes."
Centrica said: "As confirmed at our annual general meeting on 12 May 2014, the board of Centrica has been paying due attention to succession planning and has conducted extensive searches. Sam remains firmly focused on leading Centrica until his successor is in place."
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