The National Association of Pension Funds (NAPF), which represents some of the City's most influential institutional investors, said Greenbury had "not allayed concerns" among shareholders and required a number of improvements.
In evidence to the Hampel Committee on corporate governance, the NAPF criticised the way directors' performance targets were set and measured.
It also suggested that non-executive directors, who made up the remuneration committees that set boardroom pay, could benefit from specialist training to help them do the job better. And the NAPF toughened its support for a maximum rolling contract of one year for executive directors.
In a report that reflects widespread unease at the potential for abuse of the new long-term incentive plans recommended 18 months ago by the Greenbury report, the NAPF said the methods used to determine pay were often viewed with scepticism by shareholders, and confidence in board procedures needed to be bolstered.
The principles for setting performance hurdles for directors and the measurement of performance against these hurdles needed reinforcing, the NAPF told the Hampel Committee, which is the the successor to the Cadbury Committee on corporate governance. The NAPF recommended "further elaboration" of the Greenbury code on top pay, to give guidance to remuneration committees.
The Hampel Committee, led by ICI chairman Sir Ronald Hampel, should consider giving guidance to remuneration committees on how to determine directors' pay packages and how to make their reports to shareholders, which the NAPF believed should be a single, self-contained report rather than scattered through notes to the accounts.
The pension funds also recommended that all directors should be required to retire by rotation, so poor performing members of the remuneration committee could be sanctioned.
The NAPF said: "Further guidance is needed in connection with the determination of performance targets, which must be challenging without being demotivating."
The evidence to the Hampel Committee said remuneration policy should be clearly and comprehensively stated, accompanied by tables, and any new bonus arrangements or long-term incentive plans "should be explained and the measurement targets justified prior to implementation".
However, the NAPF urged that the importance of directors' remuneration issues should be kept in perspective, and any new guidelines adopted by the Hampel Committee should be flexible enough to meet the circumstances of individual companies. It also insisted that shareholder approval of individual directors' pay was impractical.
The NAPF said recent litigation had reinforced the need for rolling contracts in most cases to be for no longer than a year. Compensation has in the past been cut back in situations where directors are likely to find another job, but this "legal duty to mitigate" had been weakened by court decisions.
The NAPF also urged that company secretaries should, for the first time, be given the same security of tenure as auditors, so that only shareholders could dismiss them.
It said the Hampel Committee should resist the urge to toughen the Cadbury code with more prescriptive requirements for companies, and it should be pragmatic and flexible.
Among detailed improvements, boards should explain why any director over 70 who is up for re-election should be retained, and publish brief biographical details of all directors in the annual report.
The NAPF reaffirmed its objection to two-tier shareholdings that give one class of share fewer rights, defended the single board system against the continental two-tier structure and said there should be a pragmatic approach to implementing the Cadbury code for small firms.
The NAPF opposed the appointment of executive directors to represent particular interests, though it said all directors must take into account the interests of "relevant stakeholders" as well as shareholders.