American-born Dudley, who was appointed chief executive in the wake of the Deepwater Horizon oil spill in the Gulf of Mexico in 2010, could leave the role by this time next year, Sky News reported.
READ: Deepwater Horizon is still costing BP billions every year, but its colossal cash generation means it doesn’t really matter
Irishman Looney, who has run the FTSE 100 group’s Upstream business for as long as Dudley has been CEO, is commonly thought of as the most likely new internal candidate for CEO, along with chief financial officer Brian Gilvary.
Dudley, who was previously in charge of the oil behemoth’s Russian unit, took the reins at BP from Tony Hayward shortly after the Gulf of Mexico disaster, which has cost BP around US$70bn in fines, legal bills, compensation and clean-up costs.
In dealing with the Deepwater Horizon fallout he has sold assets and moved into new markets, while managing to continue paying a dividend for all but a few months.
Recent deals have included snapping up BHP's acquiring petroleum assets for US$10.5bn and selling its Alaska operations to Hilcorp Energy for $5.6bn.