Barclays is thought to be in discussions with a number of US institutions and private-equity investors interested in buying the iShares division, which specialises in exchange traded funds and is based at BGI's headquarters in San Francisco.
The sale is being led by Roger Jenkins, head of investment banking in the Middle East.
Barclays bosses regard £4 billion as a good price for iShares, given its size in relation to the rest of the bank.
Barclays has seen the value of the overall bank plummet from more than £40 billion to £6.8 billion since the start of the banking crisis.
BGI is one of the biggest fund managers in the world with £1 trillion on its books and iShares makes up a quarter of its business.
BGI delivered £600 million of the bank's £6 billion profit last year.