The British lender is likely to take a stake of up to 20 percent in the U.S. money manager if a deal with BlackRock does happen, the paper reported, adding that Barclays was expected to decide on who should buy BGI early this week.
Bob Diamond, president of Barclays and head of BGI, plans to join BlackRock's board if a deal happens, the paper reported.
BlackRock would have the equity and debt financing needed to do the deal, the paper said.
BlackRock Chief Executive Larry Fink went to the Middle East last week to look for funds for the deal from sovereign wealth funds, including the Kuwait Investment Authority and the Qatar Investment Authority, the paper said.
BlackRock spokeswoman Bobbie Collins declined to comment. Barclays could not be immediately reached for comment by Reuters.
A sale of the San Francisco-based unit would sideline private equity houses, which were bidding for iShares, the exchange-traded fund unit that forms part of BGI.
On Friday, a source familiar with the situation told Reuters that Barclays was still in talks with a couple of interested parties on BGI and iShares.
Barclays decided to sell iShares to buy-out house CVC for 3 billion pounds in April, but a go shop clause allows it to seek higher offers until June 18.
CVC has the right to match any rival bids for iShares -- or all of BGI -- and gets a $175 million break fee if it is left out of the deal. Bankers have said it could team up with a bigger group to buy all of BGI.
Barclays will retain a 20 percent stake in iShares if it sells it separately, and a deal for all of BGI may well be structured similarly, giving it exposure to future gains.
Barclays would be willing to sell BGI if offers approach $12 billion, bankers told Reuters last month.
Private equity firm BC Partners stopped working on its bid for iShares after Barclays seemed to want to sell all of BGI, sources told Reuters last month.
Analysts say iShares, a clear leader in both U.S. and European markets, has strong growth potential. It saw record net inflows of $89 billion last year.