With Assange facing extradition proceedings and up to five years in federal prison on the U.S. computer hacking charge, shadow home secretary Diane Abbott told BBC Radio 4's Today programme that the United Kingdom should resist handing him over.
"We do not presume guilt, of course, but we believe due process should be followed and the complainant should see justice be done", it said.
The Home Secretary, a senior Cabinet official, has some leeway to block extradition under certain specific circumstances, including cases where a person facing extradition might face capital punishment or torture in that country. The decision should be based on a number of factors, including the date of the first warrant and the severity of the charges faced.
The MPs and peers add that it is "of grave concern to us" that the Swedish authorities did not appear to have prior warning of Assange's arrest, unlike the USA authorities. Their clash prompted fears that the claim from the alleged victim had been all but forgotten.
The rape allegation has a limitation period which expires in August 2020, it adds.
"This extradition will be very hard to fight - given the nature of the UK-U.S. extradition agreement", Anthony Hanratty from the law firm BDB Pitmans told The Times.
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They added that "the United Kingdom should tell Sweden that it will have our full cooperation" if they choose to reopen the rape case against Assange.
Former Labour Lord Falconer also told the BBC that it was up to the courts to decide, not politicians.
At Thursday's hearing, judge Michael Snow described Assange as a "narcissist" and said he could consent to the extradition and "get on with your life".
Assange has denied wrongdoing.
Emily Thornberry, the shadow foreign secretary, also appeared to break ranks with Mr Corbyn and Ms Abbott, stating: "I think that what should happen is that he should be extradited to Sweden and then the Americans can make a further application to have him extradited from Sweden". Party leader Jeremy Corbyn tweeted that the U.S.is prosecuting Assange because he exposed "evidence of atrocities in Iraq and Afghanistan". The alleged victim claims that the Wikileaks founder attacked her in her sleep without a condom on despite her repeated refusals to have unprotected intercourse with him.
Two other charges of molestation and unlawful coercion had to be dropped in 2015 because time had run out.
When he took up residence inside the Ecuadorian Embassy in 2012, it was to avoid answering the sexual allegations against him in Sweden, which had sought his extradition for questioning.