Mr Gifford, 59, whose bluntness and autocratic style have made him one of the City's best-known figures, is said to have negotiated a £1m pay-off.
A spokeswoman refused to confirm or deny suggestions that Mr Gifford had planned to announce his resignation at the company's annual meeting later this month.
Nor would she comment on rumours of increasing tension between Mr Gifford and Sir Denys Henderson, the former ICI executive who takes over as chairman on 1 March.
"Mr Gifford has said all comment is to go to shareholders through the normal channels [the Stock Exchange] and not through the media," the spokeswoman said.
News that such a high-profile figure could be leaving will unsettle investors who are unlikely to be prepared to wait until the end of this month for news.
Mr Gifford was praised in the City in the 1980s for the way he cut costs and rationalised operations, but disenchantment had begun to set in about recent decisions.
The disposal of 40 per cent in Rank Xerox for £620m disappointed investors who felt the shareholding was worth more.
Mr Gifford is on a three-year contract and would not have to retire until he was 62. Although the shareholders' meeting this month will be told that all directors are to have their three-year contracts reduced to two, the change would not take effect until next year