Centrica said the appointment will be announced alongside its preliminary results on February 27 if the proposal is approved by the company’s board.
Mr Weston is currently managing director of Centrica’s US retail gas business Direct Energy.
Mr Bentley, whose leaving date has not been confirmed, is understood to be quitting over a disagreement over the company’s future strategy.
The leadership change will come amid mounting public and political anger at British Gas after it hiked energy prices for around 8.5 million households at the end of last year, while Centrica said it was on track to make profits of £1.4bn for 2012.
The group said the management changes are “part of the succession planning and the strategic evolution of the company”.
Mr Weston joined the Centrica board in July 2009 when he became the group’s North America managing director.
He was previously the managing director at British Gas’ services company, which sells boiler repair and domestic insurance.
He joined Centrica in 2001 following the acquisition of One Tel, where he was managing director of its European division.
Mr Bentley, who was brought up in Bradford, joined Centrica as group finance director 13 years ago before becoming British Gas managing director in 2007.
There has been mounting speculation over his departure after reports said he was at odds with group chief executive Sam Laidlaw, who wants British Gas to focus on new ways to promote growth in the face of creeping regulation.
Analyst Angelos Anastasiou, at Cantor Fitzgerald Research, said Mr Bentley’s departure may show “he has done as much as he can at Centrica, and is, perhaps, unlikely to be considered for the chief executive role in the foreseeable future”.
Analysts predict Centrica’s British Gas residential arm will make annual pre-tax profits of £575m, reflecting a nine per cent increase in gas consumption for the first 10 months of 2012 following cold weather.
Last year British Gas put an extra £80 on to its typical annual dual fuel bill, an average increase of six per cent which hit 8.5 million customers.
Rival npower followed with an average rise of 8.8 per cent for gas and 9.1 per cent for electricity. Both companies blamed rising costs largely outside their control, but there were fears about how the elderly and hard-up will cope with the latest rises.
The move came just months after Centrica posted a 23 per cent rise in half-year profits at its residential arm to £345m.