Aggregated News
Health officials say they have accepted that the requirement, currently in the Human Fertilisation and Embryology Bill, would hamper vital research.
The Liberal Democrats said the move was a "very sensible compromise".
But Comment on Reproductive Ethics said research interests were being allowed to override proper consent.
Scientists believe they can learn more about the development of many currently incurable illnesses by creating cloned embryos from diseased tissues.
They say the specific consent requirement would stop them using tissues already donated for general medical research by people who are dead or cannot be contacted.
If the government does drop the requirement from the bill, it is expected to insist stringent safeguards are put in place instead.
'Race against time'
Liberal Democrat health spokesman Dr Evan Harris, who has been campaigning for the change, said: "Scientists would see sometimes decades of work characterising cells wasted if one had to start again with a whole new type of cell or individual cell.
"There's a race against time to find treatments for some of these diseases and of course the job of Parliament...