The British Influence think-tank wants a judicial review of the government’s legal position on membership of the wider European Economic Area which forms the internal trading bloc.
The think-tank believes leaving that leaving the EU does not mean quitting the EEA, which extends the single market’s tariff-free trade in goods to countries including Norway, Iceland and Liechtenstein.
A legal challenge could result in Parliament being given the final say on EEA membership.
A think-tank spokesman said: “There is no need for a hard Brexit and there is no mandate for a hard Brexit.”
The group believes Britain does not need to quit the single market to control immigration, end payments to EU coffers or return powers to the UK Parliament.
It also said voters did not decide to leave the trading arrangement when they backed Brexit in the 23 June referendum.
Group chairman Peter Wilding said: “This is not stopping Brexit, this is shaping it. The country demands a win-win, smart Brexit, not a lose-lose ideological hard Brexit which will damage the UK, damage Europe and for which there is no need and no mandate.”
Fabian Picardo, chief minister of Gibraltar, said the overseas territory had a different settlement to the UK going into the EU.
He said: “I think it is going to be a differentiated deal for different sectors of industry, for different parts of the United Kingdom. It’s going to be a multi-faceted deal and one of those facets has to be the facet that applies to Gibraltar.”
Open Britain, which is campaigning to keep Britain in the single market, said polling it commissioned shows half of Leave voters are not prepared to be a penny worse off as result of quitting the EU.
Ed Miliband, a supporter of the group, said: “The Government will rightly be subject to an almighty backlash from Leave voters if it makes decisions about our economic future that make them far poorer and leaves less money for public services.”