The 58-year-old American will join from Blue Mountain Capital, the US hedge fund he joined in early 2013 after a more than three-decade career at JPMorgan.
During his time at JPMorgan, Mr Staley rose to head its investment bank, and was at one time seen as a contender to take over from Jamie Dimon as chief executive.
His appointment is likely to be widely seen as signalling a U-turn on the anti-investment bank strategy of Mr Jenkins, who one person close to the board said "did not understand investment banking".
Barclays is in the process of slashing thousands of jobs and winding down a multibillion-pound portfolio of non-core assets at its investment bank as it prepares to separate the division from its retail unit under the UK's so-called "ringfencing" law.
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The bank expanded its investment banking unit during the financial crisis by acquiring the American assets of the failed Lehman Brothers empire. But it has struggled to make a success of its challenge to the big Wall Street groups and the unit's returns remain below its cost of capital.
It is the second time Barclays has approached Mr Staley about becoming its chief executive. The first time it did so in 2012, the bank eventually balked at the high cost of buying out his contract at JPMorgan and the political risk of hiring another US investment banker to replace Bob Diamond, who was forced out by the regulator over the Libor interest rate rigging scandal.
Instead, the bank opted for the more politically acceptable choice of Mr Jenkins, who was then head of its retail bank and went on to be dubbed "Saint Antony" for his attempts to clean up the bank's scandal-plagued culture and reputation.
Mr Jenkins was abruptly fired in July by the bank's new chairman John McFarlane, who said a new "set of skills" were required and took full executive control until a permanent successor could be found.
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Mr Staley would be the second former senior JPMorgan executive to take charge of a large UK-based bank this year after Bill Winters became CEO of Standard Chartered in June.
The incoming Barclays CEO remains active in the world of banking, having joined the board of Swiss banking group UBS earlier this year. He also has international clout, as a member of an advisory committee to the Federal Reserve Bank of New York.