Mr Jenkins, who took on the role last August, said it was "only right that I bear an appropriate degree of accountability" following a year of devastating reputational blows for the group, including its £290m settlement for Libor rate-rigging.
The maximum bonus he was entitled to was £2.75m – or 250% of his £1.1m salary – but he was reportedly in line for £1m.
Downing Street has called for restraint on bankers' bonuses after former Chancellor Lord Lawson said there was a case for paying none at all at the Royal Bank of Scotland this year.
Later this month, RBS is expected to announce a bonus pool of up to £250m.
The Edinburgh bank – 82% owned by the public – is also facing a fine as high as £500m because of the Libor rate-fixing scandal. Trade unionists have claimed taxpayers are once again paying for the "greed" at the top of RBS because both sums will come from the public pocket.
Lord Lawson, who headed the Treasury from 1983 to 1989, called for the full nationalisation of RBS and launched an attack on the banking industry's bonus culture.
He said, because of the Libor scandal, there was a case for paying no bonuses at RBS and urged banks to stop fretting about losing "star performers" if bonuses were cut.
"These are not particularly impressive individuals. They're all of them easily replaced, particularly in today's labour market," said the 80-year-old Tory peer, who sits on Westminster's commission on banking standards. He added RBS should be fully nationalised and turned into a bank for increased lending to business.
The Prime Minister's spokeswoman, asked if David Cameron agreed with his Conservative colleague that RBS should not give out bonuses this year, said there was a lot of speculation on the subject, but No 10 would not comment on specific cases.
However, she urged banks to follow a policy of restraint, pointing out any bonus should reflect good performance.
Mr Jenkins, who replaced Bob Diamond last August following his resignation over the Libor scandal, said he was making his announcement to waive his bonus to avoid further unnecessary public debate.
He added it would be "wrong for me to receive a bonus for 2012" after the bank was rocked by "multiple issues of our own making".
It is still unclear, however, how much Barclays is planning to hand out in bonuses to its wider workforce and investment banking team.
Mr Jenkins's decision could set the tone for the forthcoming bonus season with the industry once more in the spotlight over its pay plans. Stephen Hester, RBS chief executive, has already waived his entitlement to a bonus for 2012, which he announced amid the bailed-out bank's computer meltdown last summer.
Yet there is still controversy building over potential payments at other banks. Next week, bank chiefs from Lloyds will be questioned by Westminsters's commission on banking standards when they are expected to be asked about bank bonuses.