It is the latest in a stream of so-called "Swissleaks" allegations that have hit the reputation of the British banking giant and caused a political storm ahead of a general election in May.
The report claims the chief executive was a client of the Swiss private banking arm accused of helping wealthy clients evade tax.
Gulliver held about $7.6 million (6.7 million euros) in 2007 in a Swiss account in the name of Worcester Equities Inc, a Panama-registered company, according to the report.
Gulliver, who is based in Britain but is domiciled in Hong Kong for legal and tax purposes, was listed as the beneficial owner of the account, the report said.
It was published on the evening before Gulliver is due to present HSBC's annual report, expected to be overshadowed by a scandal that has prompted investigations of the bank by Britain's financial watchdog and Swiss authorities.
HSBC did not immediately respond to a request for comment by AFP.
However, a representative for Gulliver told the Guardian that the chief executive had used a Swiss account to hold his bonus payments prior to 2003, when he moved from Hong Kong to London.
Gulliver's lawyers said that Hong Kong tax had been paid on this income and that his Swiss accounts had been declared to British tax authorities.
British newspapers published a letter from Gulliver apologizing for the Swiss division's behaviour in full-page advertisements last week.
Gulliver said the Swiss arm had been "completely overhauled" since 2007, when former employee Hervé Falciani stole a huge cache of data and passed it to French authorities.
British tax authority HM Revenue and Customs (HMRC) officials, accused of failing to act adequately on evidence of tax evasion in the files, are to be grilled by members of parliament on Wednesday.
In related developments, Argentina's tax authority said on Sunday that it is seeking an international arrest warrant for an investment advisor suspected of helping Argentines hide money in Swiss bank accounts.
Miguel Abadi, head of the Gems financial advisory company, was linked to secret HSBC accounts identified in the "SwissLeaks" expose published by a consortium of news organizations, including Argentina's Clarin and La Nacion newspapers.
The Argentine tax authority AFIP said in a statement that it had asked judge Maria Straccia for the "repatriation of 3.8 billion dollars in 4,040 Argentine accounts hidden in Switzerland and the international capture of Miguel Abadi."
The judge must now consider the request and relay it to Interpol.
AFIP said Abadi had used front companies and trusts in Uruguay, the Virgin Islands and Bermuda to move the funds.
The HSBC documents were among those stolen by Falciani, backing claims that HSBC's Swiss private banking arm helped clients in more than 200 countries evade taxes on accounts containing $119 billion.