In a statement ahead of the TUC annual conference today, O’Grady warned that some jobs could be at risk if the UK did not maintain access to the European single market.
As the government develops its proposals for the forthcoming Brexit negotiations, it is not yet clear if ministers will attempt to maintain full access once the UK leave the economic bloc.
O’Grady urged priority to be given to “the best deal possible for working people”.
She added: “We must build a Britain that is successful, prosperous and fair – a Britain of great jobs for everyone.
“Government must be ready to step in and work to keep the advantages we get from membership of the single market – for all of our industries, not just the City. That’s the key to a successful Brexit for working people.”
Also today, Unison urged Brexit to end the government’s austerity programme.
Ahead of speaking in a debate on the impact of the EU referendum vote on the UK economy and its public services, general secretary Dave Prentis said many people had voted in good faith to leave the EU on the basis of “false promises”.
Voters were promised more money for public services by the Brexit politicians, he stated. “When people went to cast their votes, many will have been thinking of the big red Boris bus that promised £350m a week for the NHS.
“Harsh spending cuts are pushing nurses, teaching assistants, social workers, town hall staff and those caring for our loved ones beyond breaking point.
“But the British people didn’t vote for more cuts in June, they didn’t endorse austerity, in fact they voted for the opposite – for more spending.”
As a result, Prentis urged government to “throw off the shackles of spending cuts” and invest in jobs, services and infrastructure.
He added: “A new approach from ministers would renew our country, rebalance the economy and give the next generation the opportunities they deserve.
“It could be an opportunity for ministers to turn the tables on the privatisers, the free marketeers and all those who would use departing the EU as an excuse to turn Britain into an even lower tax, lower investment, free-trade tax haven for the super-rich.”
However, he added that “we all know that won’t happen” under the Conservative government and urged the Labour party to form a united and electable opposition as an alternative government in waiting.
