One in five of those polled for the Sunday Telegraph also said they sympathised with the "feelings and motives" of the bombers who killed 52 people in attacks on the London transport system in July.
But 91% of British Muslims said they feel loyal to the United Kingdom.
Sharia is implemented to varying degrees in several Muslim countries including Iran and Saudi Arabia, where religious courts can impose punishments including stoning, amputation and execution.
In other countries Sharia is applied to specific areas such as family law, banking or religious rituals.
The poll came one day after 10,000 Muslims took to the streets of London to express their anger and hurt over the publication of controversial cartoons depicting the Prophet Muhammad.
The cartoons have been deemed blasphemous and offensive to Islam.
About 10,000 Muslims protested
in London against the cartoons
They were first published in September in a Danish newspaper and were reprinted by papers in other countries but not Britain.
The publication prompted uproar in the Muslim world, with thousands taking to the streets to protest.
Five people were killed in protests in Pakistan and 10 people were reported to have died in clashes in Libya. Sixteen died in Nigerian riots.
Islamic tradition forbids any drawings of the Prophet Muhammad in order to discourage idolatry.