Bramson's fund, Sherborne Investors, had pressured Staley since 2018 to scale back investment banking and demanded his removal over his links to U.S. financier and registered sex offender Jeffrey Epstein, but Bramson struggled to gain much traction.
The value of the Sherborne vehicle dedicated to the Barclays stake has fallen by around 150 million pounds ($209 million) compared to the 700 million pounds it raised when it was formed in 2017.
The fund, in which UK asset managers Aviva and Schroders have stakes, said it had identified another company to invest in and has begun building a position in that firm, which it did not name.
Barclays declined to comment.
"Business is not a science and so people of goodwill may, therefore, sometimes differ. In that spirit, Sherborne Investors expresses its most sincere wish that things will turn out well for Barclays, its employees, and its investors," Sherborne Investors Management said in a statement.
One top 20 investor in Barclays, who declined to be named, said a recent rally in Barclays' shares on investor bets of a strong recovery from the COVID-19 pandemic had provided Bramson with an exit route and predicted he would opt for a less complex next target.
Barclays shares have fallen roughly 15% since Sherbone first declared its stake in Barclays in March 2018.
The stock was up 2% on Friday.
'THE WAR IS OVER'
When Sherborne first invested, Barclays had been under pressure to bolster profits after Staley's aggressive push in investment banking since he joined in 2015 failed to significantly lift returns.
Barclays was Bramson's largest target to date. He engineered turnarounds at smaller firms including British private equity firm Electra after forcing his way onto the board. But a bid by Bramson for a board seat at Barclays was emphatically rejected by shareholders in 2019.
A strong showing by the investment bank helped it weather the COVID-19 pandemic, bolstering Staley's argument that it needed a diversified business to cope in a recession.
"The war is over," wrote Ian Gordon, a banking analyst at Investec.
"We sense that investor appetite for a further round of heavy restructuring charges to execute such a repositioning of the group was limited".
Bramson also urged Barclays' board to replace Staley after the UK regulator said it was examining his links with Jeffrey Epstein, but failed to win support.
Staley has said he regrets having a professional relationship with Epstein, who killed himself while awaiting trial on sex trafficking charges.
Sherborne initially invested 580 million pounds in Barclays' shares and derivatives but the investment did little to lift its fortunes.
The net asset value of the Sherborne fund is currently 79 pence per share, it said, compared to 99 pence in December 2017 before it made the Barclays investment.
(Reporting By John O'Donnell in Frankfurt and Iain Withers and Carolyn Cohn in London; editing by Rachel Armstrong and Elaine Hardcastle)
By John O'Donnell, Iain Withers and Carolyn Cohn