The 13 billion trust, which specialises in medical research and is also Britain's leading investor in hedge funds and private equity, said it had made the offer in a consortium with Blackstone Group.
A Wellcome spokesman declined to comment further. However, the consortium is understood to have submitted a bid by last Tuesday's deadline to UBS, the investment bank handling the RBS auction that is expected to fetch between 1bn and 2bn.
The confirmation signals a heating-up of the battle for the 318 RBS branches, which come with their former Williams & Glyn's name and are virtually all in England and Wales. RBS and Blackstone declined to comment.
Other committed suitors for the branches are understood to be Spanish banks Santander and BBVA, Clydesdale Bank-parent National Australia Bank (NAB) and Virgin Money.
NAB has said it is interested in being a consolidator of a banking sector shaken up by the credit crunch and bailouts that see the taxpayer holding 84 and 41 per cent stakes in RBS and Lloyds Banking Group respectively.
Santander has already taken over Abbey National, Alliance & Leicester and Bradford & Bingley's deposits and branch network business in Britain.
RBS has been ordered to sell the network by the European Commission, in return for the government aid it has received, and the sale of some Lloyds' branches is also in the pipeline.
Wellcome's entry into the fray was unexpected, as it had not previously been linked to the sale. However, the charity is thought to see the RBS branches as a long-term investment opportunity.
Keith Bowman, banking specialist at broker Hargreaves Lansdown, said: "I'm totally surprised at Wellcome, but it has been known that Blackstone has been looking for an entry into the UK banking market."
Bowman said a charity/private equity bid was unlikely to have the trade synergies available from such an acquisition as trade buyers would enjoy.
He added: "But you could argue that existing brand names are so low at present that a new player might have a real chance of promoting a new brand under new ownership very well."
Wellcome recently released figures disclosing that its annual charitable spend rose to 720 million after delivering a 5 per cent return on its investments despite the economic woes.
The group – the main source of non-governmental biomedical research in the UK – said its investments produced a gain of 580m in the year to end-September.
Set up in 1936, it is the world's second-richest medical charity, the Gates Foundation being the richest. It was launched after the death of pharmaceutical entrepreneur Sir Henry Wellcome, who left the entire share capital of his company to trustees charged with spending income to further human and animal health.