Some, like 45-year-old Tom Glover, are planning to vote for the centrist Liberal Democrats -- a minor opposition party that is very pro-European.
"I've never voted for the Lib Dems in my life, but I will this time," the computer engineer and staunch Europhile told AFP in the City, London’s business hub.
Glover was seduced by the Liberal Democrats' promise to hold a second referendum at the end of Brexit negotiations on whether to accept the terms of the deal, or reject them and stay in the bloc.
"The leader (Tim Farron) isn't the most charismatic in my opinion but the idea that they would try their best to stay in Europe is enough for me to vote for them," he added.
But few think like Glover and, despite unexpectedly receiving the backing of the respected weekly magazine The Economist, the Liberal Democrats continue to languish in the polls.
The party only has seven percent of voting intentions in a recent poll by YouGov -- which interviewed 1,875 people on May 30 and 31.
That was well behind the ruling Conservative Party on 42 percent and the main opposition Labour Party on 39 percent.
'Best for the UK'
Long-time supporter Benjamin, 31, said he would not vote for the Lib Dems despite being anxious over Brexit as his partner is European and his industry is dependent on business with Europe.
"It's just that they are not going to win. There is no point," he told AFP.
For the pension fund employee, it is crucial to block Prime Minister Theresa May from leading the Brexit negotiations.
"I don't believe she or the Conservatives are doing what is best for the UK," he said, arguing that "it would be better to have a Labour government."
According to YouGov, 53 percent of "Remainers" -- those who voted to stay in the bloc -- intend to vote for the main opposition party led by Jeremy Corbyn, even though his campaign to stay in the EU was lacklustre and he is now promising Brexit.
Forty-eight percent of them believe "the government has a duty to carry out" Brexit, according to a YouGov poll from May 8.
"People voted, so at the end of the day, if you go back on this referendum, why would you abide by any other referendum?" Natasha, a human resource professional told AFP.
Describing herself as "unrepresented" on the campaign trail, Natasha argued that the referendum's close result "gives credence to the fact that there should be a soft Brexit" where Britain would remain inside the European single market and allow immigration.
It is for these undecided voters that Gina Miller launched her "Best for Britain" initiative within hours of May taking the country by surprise in April by announcing a snap election.
Her crowd-funding appeal promised to put Brexit at the heart of the campaign and collected more than £400,000 ($514,000, 457,000 euros).
The investment fund manager has distributed the money to 25 candidates from either Labour, Lib Dems, the Green Party or independent.
All pledged to fight against the tougher Brexit advocated by the prime minister.
For Miller, this "progressive alliance" may be the answer to what she see as a "vacuum" in British politics.
"The most effective thing to do is tactical voting," she told AFP.
Voters, she argued, should cast their ballot "for the character of the individual, not the character of the party".