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"The ball is very much in the court of some of their activists," a spokesman said.
But Greenpeace responded by repeating its belief that GM crop trials were not in the best interests of the public.
The president of the British Association for the Advancement of Science, Sir Richard Sykes, called for the sites to be kept secret at Britain's Festival of Science in Sheffield.
Sir Richard said this would enable scientists to collect the data without disruption.
But Greenpeace campaign director, Dr Douglas Parr, said: "If science is to progress and serve the best interests of the public, the industry should redirect the public money it is ploughing into this unwanted, unpredictable technology into research into sustainable organic agriculture.
"It is not a question of 'pro' or 'anti-science', but a question as to which scientific research will benefit mankind in future generations."
At the moment, protesters can find precise details of the locations in the press and on the Internet.
Cabinet Office Minister Jack Cunningham told BBC Radio 4's Today programme that could change if GM crops continued to be targeted.
He said: "We cannot allow a situation to continue where premeditated vandalism and physical destruction of other people's property prevents us building up the scientific knowledge we need to make properly informed decisions.
"If we cannot proceed with the test growing in an open way, then we shall have to consider alternatives."
'Cutting edge'
Mr Cunningham said he did not think that position had been reached yet, but said he was monitoring the situation and looking at what happened in other countries.
Sir Richard, who is also chairman of the biotechnology giant Glaxo-Wellcome, called for future testing to be secret to avoid the "vandalism" that has destroyed a number of experimental crops in the UK.
He said it was certain that GM technology would continue to be developed elsewhere and its full potential and rewards would be realised by the UK's competitors.
"GM crops are at the cutting edge of agriculture and when we start to lose this high technology and the scientists, we start to lose part of our knowledge-based economy," he said.
"That is certainly detrimental to the UK."
Sir Richard also said that UK companies working in GM technology are already not getting investment because of concerns over the UK's attitude and are therefore not able to do their research.
Britain has offered to host a conference on behalf of the 29 nations in the Organisation for Economic Co-operation and Development next March.
Dr Cunningham said the conference should make a "significant contribution to the GM debate".