The Labour leader said his party were not in favour of another vote at the moment - but wouldn't be drawn on whether he would stick to it in future.
He said: "Once again the very top of Labour, this time Jeremy Corbyn and Emily Thornberry, have failed to rule out a second referendum".
Mr Corbyn however appeared to finally settle on whether his party would call on the British Government to stay in the single market, saying Britain can not remain a member of the single market because of its decision to leave the European Union.
Mr Sturgeon lashed out at the Labour leader on Twitter, arguing he was misleading people by arguing Brexit meant leaving the single market.
"The goal of the Withdrawal Bill is to provide a functioning statute book on the day we leave the European Union, not to pursue policy choices like this".
Is that what we have said to the European Union in opposition? Yes.
When it was put to Mr Corbyn that he was not saying he would never support another referendum, the Labour leader said: "We are not calling for one either".
"At the moment... We proceed in good faith".
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Tom Cotton, R-Ark., and David Perdue, R-Ga. - said Friday that they " do not recall " Trump "saying these comments specifically". Trump then made "vile and vulgar comments " about African nations, Durbin said , calling them "shitholes".
"So maybe just maybe we should have a second referendum on European Union membership".
Nigel Farage surprised both Remain and Leave supporting politicians on Thursday when he said he could back a second referendum in order to secure the Brexit vote.
One challenged him and said "let us leave". However the survey said only 18% of Tory voters want a second poll and 79% were against.
"This amendment would not keep the United Kingdom in the single market and customs union, nor does it intend to deal with the serious flaws in this Bill", they said.
Scottish First Minister Nicola Sturgeon insisted that staying in the Market and Customs Union would be the "least damaging" option post-Brexit - and called on the Labour leader to get behind it.
The comments came after Labour MP Wes Streeting said the party was now the "single biggest barrier" to Britain remaining in the single market.
The SNP, Liberal Democrats, Plaid Cymru and the Greens will collectively back an amendment to the EU Withdrawal Bill this week.
Single market membership after Brexit has become a divisive issue within Labour, with members struggling to reach a common party line on the matter.