The switch by Hampton to the top role at Britain's biggest pharmaceutical company is expected to be confirmed next month.
He would replace Sir Christopher Gent, the current chairman, who has indicated he intends to stand down at the end of 2015 after almost 10 years in the role at the pharmaceutical company, which is battling for its reputation in the midst of bribery allegations.
The RBS chairman has been facing questions about his intentions to remain at the 81% taxpayer-owned bank since Stephen Hester quit as chief executive in June 2012. At the time Hampton said he had "no imminent plans" to quit until Hester's replacement had "their feet under the table".
Hester's successor Ross McEwan has now been at the helm of the bank for a year, while Hampton's remarks that a chairman should serve between five and seven years have been interpreted as meaning he would seek to leave after the bank's annual meeting in 2015.
If he does take the chairman's seat at GSK, which Sky News reported would be confirmed next month, he will joining a business facing severe reputational challenges.
The trial is due to start on Friday in Shanghai of the private investigators brought in to investigate a covert sex tape of the former head the pharmaceutical company's China division.
The secretly filmed video of Mark Reilly and his Chinese girlfriend in his Shanghai flat was sent to GSK executives while Reilly was accused by the local authorities of running a bribery network. The investigators brought in by GSK – Peter Humphrey, a Briton, and his wife Yu Yingzeng, an American national – are now facing trial for illegally purchasing information about Chinese nationals.
The allegations of bribery sparked the GSK chief executive Andrew Witty to declare the company had a "zero tolerance" approach to any attempts at corrupt sales practices of the type it faces in China and other countries in which it operates. However, GSK also faces of bribing doctors in Poland, Iraq, Jordan and Lebanon. In May, the Serious Fraud Office said it had launched a formal criminal investigation into GlaxoSmithKline's sales practices.
For Hampton to take up the chairman's role at GSK he would likely be appointed to the board as non-executive director first with his elevation to the chairman's seat at a later point in 2015.
A former corporate financier, Hampton has held a number of roles in the boardrooms of major companies including finance director of British Steel and a short stint at Lloyds before the 2008 banking crisis. He was chairman of supermarket group J Sainsbury when he was parachuted in to chair RBS after its £20bn taxpayer bailout.
The government is yet to sell off any of its stake in RBS and had indicated that it does intend to do so before May 2015, unlike Lloyds Banking Group where a 43% stake has been cut to 24%. Hampton last month oversaw the release of RBS's interim results a week early because of better than expected profits of £2.6bn – the highest since the 2008 bail out.
Hester, who left RBS a year ago, joined troubled insurer RSA in February and on Thursday reported a return to profitability after being hit by problems its Irish arm.
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