Mr Weston, a former soldier, has been Centrica's managing director for international downstream since last year, in a role that encompasses Scottish Gas and British Gas's residential businesses.
His possible move comes after the exit of Rupert Soames, the grandson of wartime prime minister Sir Winston Churchill to become chief executive of outsourcing company Serco.
If Mr Weston does secure the role at Aggreko, which has a large manufacturing plant in Dumbarton, his departure is expected to be announced at roughly the same time as that of Sam Laidlaw, Centrica's long-serving chief executive.
Aggreko's interim chief executive is Angus Cockburn who worked with Mr Soames as finance director of the Glasgow-based temporary power supplier for more than a decade.
There have been reports Mr Soames wants to bring him to Serco.
Mr Weston was previously managing director of Centrica's North American business, having run British Gas Services and British Gas Business in the past.
He joined Centrica in November 2001, following the acquisition of One Tel where he was the managing director for Europe.
Previously, he worked for Cable & Wireless and also spent seven years in the army with the Royal Artillery.