Prime Minister Theresa May still expects the election can be avoided by Britain by leaving the EU.
Labour would become the largest party in the House of Commons with prominent Tories, including Work and Pensions Secretary Amber Rudd and arch Brexiteer Iain Duncan Smith at "high risk" of losing their seats.
Talks between the government and Labour are set to continue over the Easter parliamentary recess in the hope of finding an agreement that will be acceptable to MPs.
He warned the talks could not be allowed to drag out for months, but showed little indication the government may drop its requirement that future EU ties cannot involve a customs union.
Despite the best efforts by Remainers to use European Parliament elections to their advantage, think tank Open Europe said last week that the polls are likely to result in a "protest" vote, where Britons will "send a message" to the establishment, punishing them for failing to deliver Brexit.
Speaking on BBC Radio 4's The World This Weekend, Lord Falconer said he feared that the time the commission would take to complete its investigation would be "too long" for Labour to take effective measures against the reputational threat it faced now.
"I don't want to compromise what is at the moment a space where we are testing with the opposition, and they are testing with us, particular ways in which we could move forward".
He said meetings later this week would be divided into different subjects led by relevant ministers and their Labour opposite numbers.
This morning, Richard Corbett, the leader of the party's 20 MEPs, said failure to clearly support a second referendum could see the party losing millions of young voters to rivals with a clearer message.
"If Labour does not reconfirm its support for a confirmatory public vote on any Brexit deal in its manifesto, then it will hemorrhage votes to parties who do have a clear message".
"If, on the other hand, we do offer clarity and a confirmatory ballot, we could do very well".
Britain was due to have left the EU earlier European elections in May.
Last week the European Council extended Britain's membership of the bloc until October 31 to give more time for the British parliament to agree on a withdrawal deal.