Mr Corbyn also said today that there would "have to be a customs union with the EU obviously" but again insisted that Britain can not be a part of the EU's Single Market without being a member of the EU.
The SNP, Liberal Democrats, Plaid Cymru and the Greens will collectively back an amendment to the EU Withdrawal Bill this week.
MPs such as Chuka Umunna, Stephen Kinnock and Heidi Alexander have all urged the party leadership to embrace a softer form of Brexit.
The SNP, Liberal Democrats, Plaid Cymru and the Greens are set to back a separate amendment this week that would prevent ministers being able to take the United Kingdom out of the single market and customs union without the approval of parliament, but a Labour spokesman said over the weekend the party did not intend to support it.
In a speech to the Fabian Society's annual conference in London on Saturday, Streeting will say: "On the UK's future relationship with the European Union - the single biggest issue facing our country in a generation - we find ourselves in the bad position where it is the Labour party that now stands as the single biggest barrier to the UK's membership of the single market and customs union".
"What happens this year in Parliament will have huge and lasting repercussions for the country - which is why we're putting party differences aside and working alongside each other".
"But at the moment, and as things now stand, we proceed in good faith, we do as we are instructed".
Both Labour and the Conservatives have said the 2016 vote to leave the European Union should be honoured.
He added: "We've had a Referendum and we lost and you won, I accept that" but he said the public was becoming more anxious about the issue.
"Every step of the way Labour are trying to frustrate the Brexit process rather than make a success of it".
Labour's formal position is that Britain's interests would be best served by retaining membership of the single market and customs union during the transition period that will begin in 2019.
Mr Corbyn said his party was not seeking a new Brexit vote at the moment but failed to explain whether they would in the future.
'There are also aspects of the single market one wants to think about such as the restrictions on state aid to industry, which is something that I would wish to challenge'.
But he expressed reservations about the European Union single market, which allows for goods to be traded freely between European Union members.
Sir Keir said he wanted the same trade outcomes as backbench critics calling for the party to campaign to stay in the single market and customs union.
Labour has been strongly critical of the U.S. president and the decision to invite him on a state visit to the UK.
"The US one is obviously culturally and economically significant and important".
"We have to make a special relationship with the European Union".
The First minister accused Mr Corbyn of seeking to "mislead people" by persisting with "inaccurate" comments about the trading bloc.