The Labour leader insisted that "the single market is dependent on membership of the EU" and remaining in the single market would not respect the outcome of the 2016 referendum.
Streeting is one of a well-coordinated group of vocal backbench rebels campaigning for leader Jeremy Corbyn to harden Labour policy by saying he would back continued membership of the single market and the customs union.
"As for a second referendum.in either case... if 90 per cent of the population were saying we should stay in the European Union, that would be a challenge that would be there for all of us who are democrats", she said.
"With Labour, there would be a majority in the Commons for single market membership, but not without us".More news: Mario Tennis Aces Announced for Nintendo Switch
The Labour leader told ITV's Peston On Sunday: "We are not supporting or calling for a second referendum".
"We have to make a special relationship with the European Union".
"As for a second referendum, the question is are you talking about the divorce or the final relationship, because the final relationship will be agreed in many years' time", she said.
"But at the moment, and as things now stand, we proceed in good faith, we do as we are instructed".
"Only the Conservatives are working to ensure that when the United Kingdom leaves the EU we take back control of our laws, money and borders - Labour just can't be trusted to deliver on leaving the European Union".More news: Wenger struggles for answers after "very frustrating" Bournemouth loss
Alastair Campbell believes there will be a "fresh referendum" and/or a general election but said that he now believed "Brexit will happen and it will be a disaster".
Among Labour voters 64% are in favour of a second referendum with 31% against. However the survey said only 18% of Tory voters want a second poll and 79% were against. "It's not a game of tic-tac-toe - we're not here to have the best of three", he said.
Pro-Remain Labour Chuka Umunna also questioned the statement, pointing out countries like Norway and Iceland maintained access to the single market despite being outside the European Union.
The First Minister called on the Labour leader to get off the fence on the key issue, claiming there would be a Commons majority for staying in the single market if he threw his weight behind the plan.
But the comments sparked a swift backlash from some Labour MPs.More news: Dark Souls Remastered publisher pegs leaked 50% loyalty scheme as a mistake
"I don't quite understand why she would keep saying, "join the single market" when leaving the European Union means you leave the single market", the Labour leader said in response to Ms Sturgeon's appeal.
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