According to Sky News, Barclays is one of a small number of bidders that have submitted offers to buy the business, which is currently owned by Toronto Dominion (TD) Bank.
If successful, Barclays would likely seek to merge TD Direct, which has around 200,000 customers, with Barclays Stockbroking.
Barclays is far from certain to emerge as the final buyer, according to sources who also cautioned that TD Bank may yet decide to halt the sale process entirely.
Potential job losses are understood to be a significant hurdle to any sale.
TD Direct also operates an international arm based in Luxembourg, which a market source told Sky News had prompted some unease inside Barclays following a series of fines from regulators in recent years related to dealings with offshore clients.
Largest UK brokers
TD Bank entered the UK share dealing market with the opening of Green Line Investor Services in 1997, and has owned the entirety of TD Direct since 2010, when it acquired the remaining 25% of the business from BGL BNP Paribas in France.
A string of takeovers and a joint venture with NatWest Stockbrokers in 2002 helped to build TD Direct into one of the UK’s largest brokers, and it now has roughly £13bn ($18.5bn, €17.1bn) of customer assets under administration.
Trail Hargreaves Lansdown
Barclays Stockbrokers manages an asset pool of around £15bn, meaning that a combined group would still trail behind the much larger Hargreaves Lansdown.
When contacted by Sky News a TD Bank Group spokesman said it did not “comment on rumour or speculation”, while Barclays also declined to comment.