This comes as the bank grapples with an increasing bill for mis-selling compensation and a £5.8 billion fundraising drive. Jenkins has previously warned that it could take up to a decade for the bank to win back customers' trust.
The Barclays boss said he "respectfully declined" a bonus for 2013 given "legacy litigation and conduct issues" at the bank.
In a statement, he said: "2013 has been a year of considerable positive change for Barclays, and I am particularly proud of the progress we have made in starting to rebuild trust, in defining and implementing a common culture, in repositioning the business for the future, and in significantly improving our balance sheet.
"While all of these actions are in the long-term interests of our shareholders, I am aware of the very significant costs which have been required to address legacy litigation and conduct issues in 2013, as well as to exit assets and businesses we no longer wish to participate in.
"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."
The exact bonus Jenkins chose to waive is not clear, but he is entitled to one as high as £2.7 million.
The news about Barclays comes as its banking rival Lloyds has put aside a further £1.8 billion in compensation for mis-selling payment protection insurance, taking the total bill to £10 billion.