The deal could see Barclays take a stake of up to 20 percent in the enlarged asset manager, according to the sources, who asked not to be named as the talks are confidential.
It sounds like a deal is coming and the price is undeniably attractive, said Ian Gordon, analyst at Exane BNP Paribas.
It's an opportunity to realize a premium price for a business that has delivered exceptional growth but might be expected to grow at a slower pace from here, and also bolster capital ratios, he added.
Barclays said on Monday it had received proposals for BGI and iShares from a number of parties, including BlackRock, and was continuing talks. Blackrock also confirmed the talks but both companies said issues remained that could derail a deal.
San Francisco-based BGI is the world's biggest fund manager with $1.5 trillion in assets under management and would more than double the size of BlackRock.
BGI is staffed by academics from a range of disciplines from economics to engineering, attracting graduates from the nearby University of California at Berkeley, many of whom would get a windfall if a deal is reached.
Barclays agreed to sell iShares, which is part of BGI, to buy-out house CVC for $4.4 billion in April, but a go shop clause allows it to seek higher offers by June 18.
By 0820 GMT Barclays shares were down 2.2 percent at 279 pence, underperforming a weaker European bank sector, and valuing the bank at just under 24 billion pounds ($38 billion).
Bankers told Reuters last month that Barclays would likely sell BGI if offers approached $12 billion.
BlackRock is likely to get funding for a deal from Middle East investors, possibly including some Barclays shareholders, according to media reports.
The Qatar Investment Authority and Adia, the government investment arm of Abu Dhabi, are in talks alongside Kuwait's KIO to inject $3 billion in BlackRock for a 12 percent stake, the UK's Sunday Telegraph newspaper said.
BlackRock had $1.3 trillion in assets under management at the end of March. Its market value was just over $21 billion at Friday's close. Bank of America
QIA bought shares in Barclays last year and could own a stake of over 12 percent if capital instruments and warrants are converted.
It said last week it remained a supportive shareholder of the bank, after an Abu Dhabi fund made a $2.5 billion profit from selling a stake.
Barclays is likely to get two seats on BlackRock's board if it completes a deal, one of which is likely to go to Bob Diamond, president of the bank and head of BGI, the Telegraph said.
A deal would land millions for Diamond and hundreds of BGI employees due to a lucrative equity ownership plan dating back to 2000, that could leave staff owning 10 percent of BGI worth over $1 billion on paper.
An offer of $12 billion would represent 12.8 times last year's pretax profit and 11.3 times earnings before interest, tax, depreciation and amortization (EBITDA), above the 10.1 times EBITDA valuation on CVC's bid for iShares.
A sale of BGI would further boost Barclays' capital. Its core Tier 1 ratio will rise to 7.2 percent under the planned iShares sale, and selling BGI would lift the ratio to around 8 percent, analysts estimated.
CVC would have five days to match any rival offer for BGI or iShares and gets a $175 million break fee if it is left out of the deal.
(Editing by Greg Mahlich)