The leading business figure, who is a former chairman of supermarket chain Sainsbury’s, is due to be confirmed as Glaxo’s chariman-designate before the weekend, Sky News reported.
He has been at the forefront of RBS’s turnaround efforts after taking the role of chairman in early 2009 alongside chief executive Stephen Hester.
Sir Philip is expected to take over at Glaxo from Sir Christopher Gent in the middle of next year, while RBS has reportedly already engaged with headhunters about finding its next chairman.
His planned departure from the RBS board comes with the Government still a long way from returning the bank to private hands.
Drugs company Glaxo has been in the spotlight recently after it was fined £297 million and its former country manager handed a suspended prison sentence in China for bribery.
Sir Philip has been finance director at Lloyds TSB, BT, British Gas and British Steel and is a former chairman of UK Financial Investments, which manages the UK Government’s shareholdings in banks.