Barclays, which is still looking to repair its reputation following its role in the Libor interest rate-rigging scandal, is among companies under investigation for possible manipulation of foreign exchange trading.
Bonuses for bankers meanwhile remain a politically sensitive issue in Britain, six years after the financial crisis that led to huge bailouts of some of the country's biggest banks, namely RBS and Lloyds.
In Thursday's statement, Jenkins said 2013 was a good year for Barclays and that he was "particularly proud of the progress we have made in starting to rebuild trust."
"When combined with the substantial rights issue we completed in the autumn, I have concluded that it would not be right, in the circumstances, for me to accept a bonus for 2013, and I have therefore respectfully declined the one offered to me by the board," Jenkins added.
Prime Minister David Cameron last month announced that the government would cap cash bonuses at the state-rescued Royal Bank of Scotland (RBS).
