The former prime minister said the Government faced negotiations of "unparalleled complexity" if it was to achieve an agreement that replicates as closely as possible Britain's existing trade arrangements with the EU.
While voters backed Brexit in last year's referendum, he said he believed it was "possible" the public mood would change if it did not result in the promised benefits, and Labour should be ready to capitalise on that.
"A few weeks ago in the House of Commons [Brexit Secretary] David Davis said they were going to deliver a deal with exactly the same benefits we now have from the single market and the customs union. We should hold them to that," he told BBC1's The Andrew Marr Show.
"If they are going to try and deliver exactly the same benefits as we have now in the single market and customs union, this is an endeavour of unparalleled complexity.
"When the only thing people can point to is, you are going to control a section of this EU immigration as the reason why we want to do this, I think it is possible - I don't put it higher than that - that people start to think, is this really the thing that is going to be important.
"What the Labour Party should say is, 'We believed in Remain, we still think the best thing is for Britain to be part of the European Union, we acknowledge the people voted against that, we acknowledge therefore the Government have a mandate to negotiate Brexit, but we are going to hold them to the test that they have set, and if they do not pass that test then we are going to retain the right to represent the people of this country should their will change, to offer them the option of staying'."