(Adds Cameron comments, Bhatia)
By Kylie MacLellan and Elizabeth Piper
LONDON, April 5 (Reuters) - British Prime Minister David Cameron, under pressure after his late father was named in the "Panama Papers" revelations over tax havens, said on Tuesday he did not own any shares or have any offshore funds.
In remarks during a visit to a business in central England, however, he did not answer a question on whether he or his family had benefitted from offshore investment funds set up by his father. On Monday, his spokeswoman said it was a "private matter"..
Cameron's comments came as the leader of Britain's main opposition party called on the government to tackle tax havens, accusing Cameron of allowing "the super rich elite dodge their taxes".
Cameron has been urged to clamp down on tax evasion and avoidance in British-linked territories after leaked documents from a Panamanian law firm alleged the world's rich and powerful used secretive offshore company structures to stash their wealth.
The documents named his late father, Ian, and members of Cameron's Conservative Party among the list of clients who used the law firm, Mossack Fonseca.
The government has promised to investigate the leaked data but opposition Labour leader Jeremy Corbyn called for more to be done, including setting up an independent investigation.
"There cannot be one set of tax rules for the wealthy elite and another for the rest of us," Corbyn said at the launch of Labour's campaign for local elections next month.
"The unfairness and abuse must stop ... no more lip service, the richest must pay their way."
Corbyn said Britain had a "huge responsibility" as many tax havens are British overseas territories, like the British Virgin Islands and Cayman Islands, or Crown Dependencies, such as Jersey or the Isle of Man.
When Britain hosted the G8 summit in 2013, Cameron put tackling tax avoidance at the heart of the agenda. Some of Britain's former colonies increasingly rely on a combination of tourism and revenues from shell companies and trusts which often hide wealth.
Three years later, some opposition lawmakers say the release of the "Panama Papers" show the battle is far from won and are demanding that Cameron stamp British control over its overseas territories, most of which are self-governing.
According to media which have seen Mossack Fonseca's files, more than half of the 200,000 companies set up by the firm were registered in the British Virgin Islands, where details of ownership do not have to be filed with the authorities.
The Financial Secrecy Index puts the British Virgin Islands 21st out of 92 countries ranked according to their secrecy and scale of their offshore financial activities.
The head of the Global Forum on Transparency and Exchange of Information for Tax Purposes secretariat, Monica Bhatia, said the British Virgin islands had made improvements in transparency over the last five years, in contrast to Panama.
Dominic Grieve, a former attorney general and Conservative chair of the Intelligence and Security Committee, said preventing overseas territories from running their own financial services would push criminality elsewhere.
"The government has a responsibility towards encouraging overseas territories to find legitimate ways of economic development and the financial sector is undoubtedly such a legitimate method," he told BBC radio, adding that people only use tax havens if their own systems were "onerous".
"The best way of ensuring that ... people do not want to go to the BVI is to provide the right environment domestically." ($1 = 0.7032 pounds) (Additional reporting by William James and Leigh Thomas,; Editing by Jeremy Gaunt)