Hiring an investment banker is being seen by some analysts as a prelude to making a push back into that industry, one which the last holder of the top job, Antony Jenkins, was backing away from.
But will an expansion into a sector dominated by a few American names be successful? After all, this won't be Barclays first drive into investment banking.
During the deregulation of the City of London in the 1980s, Barclays bought stockbrokers De Zoete & Bevan, and Wedd Durlacher, a firm of stock market wholesalers.
The resulting amalgamation was BZW. But its expansion was curtailed by then-chief executive Martin Taylor, who broke it up, selling its equity business, and retaining fixed income as a new company, Barclays Capital.
He told investors it wasn't profitable, and the capital it sat on could be better used.
The second expansion came under the leadership of a more recent figure at the bank: Bob Diamond. Starting at Barclays in 1996, he expanded its fixed income (bonds) and foreign exchange businesses. Barclays Capital became the lender's most profitable unit.
But after the Libor scandal and his subsequent sacking, retail banker Antony Jenkins was promoted to chief executive and began cutting trading desks and the investment bank's salary bill.
Mr Staley's posited arrival was not met jubilantly by investors. Barclays stock fell by as much as 3% following the news. A cloud around the way the industry is viewed in the UK may the reason, says Chirantan Barua, banking analyst at Bernstein.
"Every time someone goes up on stage and says we are going to be a retail bank, the stock goes up," he says.
"Because we hate investment banking. The other thing that happens is, to be profitable in investment banking you need scale and therefore capital.
"Where are you going to get that from? That's why the reaction is negative.
"Getting Jes in would mean moving the bank more towards an investment bank."
For an enlarged investment bank to work, it will have to be big enough to take on competitors across the Atlantic.
Big clients like hedge funds are less likely to trade with a bank that won't do complex lending, and picks and chooses business, Mr Barua says.
"Having just the juicy part - that's not how it works."
He estimates Barclays may need to find Â£4-5bn of extra capital, through raising it from investors or selling something.
There are two alternative strategies for the investment bank, however.
"They should really just carve out the investment bank and ship it back to the US where they bought most of it," says Cenkos Securities analyst Sandy Chen, referring to Mr Diamond's purchase of Lehman Brothers's north American assets, which be bought after the bank collapsed.
US investment banks like Goldman Sachs and Morgan Stanley have a competitive advantage over their European peers in the form of friendlier regulators.
What the detached investment bank would lose is the strength the larger group can provide in the form of customer deposits.
This wad of cash - liquidity strength - allows the group to borrow cheaply in the markets and fund the investment bank. But competitors such as Goldman seem to cope without such advantages, says Mr Chen.
The other option was hinted at last week by Barclays chairman John McFarlane, who told the Financial Times that European investment banks should consider merging to create a regional champion to compete with US rivals.
"Barclays could come up with a joint-venture scheme and then it could be spun out," says Christopher Wheeler, an analyst at Atlantic Equities.
He adds that candidates with whom to merge the business could include a Swiss giant such as Credit Suisse or UBS, a global operator such as HSBC, or a French competitor such as Societe Generale, which has little overlap with Barclays in investment banking.
This would be a less expensive option than finding the capital to bolster a Barclays-owned firm, he says, and would allow a new chief executive such as Mr Staley to do all the other jobs that will be demanded of him.
"He's got to see them through ring fencing," Mr Wheeler adds in reference to the proposed process of separating retail and investment banking operations.
"The job he has is to reinvigorate the investment bank. It's not collapsed by any means - it's still a serious player, but you have to pull the whole thing together," says Mr Wheeler.
Mr Staley will still need to be approved by regulators as a choice for Barclays. If they say yes, it may be a sign things have thawed between them.
Investment banking is a "complex business", so it's hard not to have an investment banker running a bank that owns one, Mr Wheeler says. But after Mr Diamond's departure, it would have been very hard for Barclays to have chosen anyone other than a retail banker like Mr Jenkins.
Mr Staley's hiring is "not a foregone conclusion" says Mr Chen. Regulators may challenge Mr Staley's experience in retail banking, he adds.
Barclays said: "The process of appointing a new group chief executive officer has not yet concluded, and Barclays will provide a further update once that is complete."
Whatever happens, investors will want to see a more concrete plan for the investment bank, says James Box, an analyst at fund manager Brewin Dolphin, "especially when we are looking at how businesses like this need huge scale to succeed".
And executives at the bank will have to answer the question which perennially hangs over investment banks they own - for whom it will make money.
"If you want to make money from investment banking, it's best to work in one, not hold shares in one," says Mr Box.
Barclays is close to appointing former JP Morgan banker James "Jes" Staley as its new chief executive.