And it is threatening legal action if either Vote Leave or Grassroots Out are chosen instead.
The party says neither group can speak for anti-austerity campaigners who want to leave the EU, due to their "pro-business" and "reactionary" views.
A decision will be made by the Electoral Commission on 14 April.
The Commission can select one designated lead campaign for both the "Leave" and "Remain" sides ahead of the referendum on EU membership on 23 June.
The watchdog will judge each applicant's merits on the basis of a range of criteria, such as level of cross-party support, campaign tactics and organisational capacity.
The chosen campaigns will get access to a grant of up to Â£600,000, an overall spending limit of Â£7m, campaign broadcasts, free mailshots and free access to meeting rooms.
On the Leave side, it had been expected to be a straight fight between two groups - Vote Leave, which is backed by London Mayor Boris Johnson, cabinet ministers including Michael Gove, UKIP MP Douglas Carswell - and Grassroots Out, which is supported by Tory MPs Peter Bone and Tom Pursglove, UKIP leader Nigel Farage and Labour MP Kate Hoey among others.
But in a surprise move the TUSC has joined the fray by applying to the Electoral Commission for selection as the official Leave campaign.
Only one group, Britain Stronger in Europe, has applied to be the lead Remain campaign.
The TUSC claims the other exit groups cannot meet the test set down by the 2000 Political Parties and Representation Act requiring referendum campaigners to "adequately represent" all those campaigning for a desired outcome.
"We want the Electoral Commission to recognise the reality that while we can't claim to represent the hedge funds which fund UKIP and the Tories, they can't claim to represent the anti-austerity millions who want their voices to be heard," said Clive Heemskerk, the TUSC's national agent.
Choosing either Vote Leave or Grassroots Out to carry the flag for the Leave campaign would be a "political decision", Mr Heemskerk said, which would exclude all those arguing for "an exit left" strategy and would give the Remain campaign a "five to ten point boost".
While admitting the TUSC's original intention was to call for neither group to be chosen, he said it had submitted a "serious application" after concluding its concerns would not be addressed.
The TUSC was only set up in 2010 - but Mr Heemskerk said its anti-EU stance was rooted in the left's historic opposition to the Union as a vehicle for corporate interests. He dismissed Labour supporters of Vote Leave and Grassroots Out as "pro-business Blarites".
The TUSC application is being fronted by ex-MP Dave Nellist, a former backbench ally of Labour leader Jeremy Corbyn.
Mr Nellist was deselected as a Labour candidate in 1992 and later expelled from the party for his links to the Trotskyist group Militant.
Mr Heemskerk said Mr Corbyn's personal scepticism about the EU was well known but Labour's official policy of campaigning to remain in had been "dictated" by those on the centre and centre-right of the party.
"The socialist voice against the EU has been muzzled," he claimed. "It is a tragedy that Tony Benn and Bob Crow died two years ago. They would have been crystal clear."
The TUC, the main umbrella group for Britain's trade unions, backs staying in the EU although some unions are uneasy about Labour's role in the campaign so far and some have decided to remain neutral.
Rail union the RMT, which is not affiliated to Labour and which has funded its own anti-EU party in the past, is bankrolling the TUSC bid to for official designation.
Mr Heemskerk said other unions were considering their stance and Unison, on whose executive committee TUSC has strong representation, could support its call to not cooperate with the Leave campaign.
Asked whether TUSC had the organisational capacity to run a referendum campaign, he said it put up more than 130 candidates at last year's general election and joked that the RMT had shown itself able to "bring London to a standstill" in past Tube strike action.
He also warned the party would not rule anything out if "its arguments were not listened to".
"If we don't get a meeting with the Electoral Commission, then there is a strong case for a judicial review."
Vote Leave is backed by a number of Labour MPs, including Gisela Stuart and Labour-supporting businessman John Mills while Grassroots Out's supporters include Labour's Kate Hoey and former Respect MP George Galloway.
Grassroots Out welcomed TUSC's intervention and said it hoped it would play a role in the campaign, whoever won the designation.
"Trade Union and Socialist Coalition has terrific reach across the country and it's great to have workers' voices so well represented," said Brendan Chilton, general secretary of Labour Go.
"We look forward to working with them towards our common goal to free the UK from the shackles of the EU."
The Electoral Commission said it had the option to meet the representatives of the different groups to clarify points or seek further information.
"The Commission will make its decision on designation of lead campaigners, as soon as practicable within the period from 1 to 14 April," it said in a statement.
"The Commission will also publish on its website information about how the decision was reached, including the application forms submitted by campaigners," it said.
The Trade Union and Socialist Coalition has launched a bid to be designated as the official campaign to get Britain out of the European Union.