A letter written to Ms Abbott, signed by more than 70 MPs, called for Mr Assange to be extradited to Sweden so that the "formal investigation into an allegation of rape can be concluded".
On Friday evening, shadow foreign secretary Emily Thornberry said Assange should be extradited to Sweden before any attempt to get him to the US.
Assange was arrested on Thursday by British police after spending seven years holed inside the embassy.
Assange sought refuge in the Ecuador embassy, in Knightsbridge, west London, in 2012 to avoid extradition to Sweden over allegations of sexual assault.
Elisabeth Massi Fritz, the lawyer for the woman who reported being raped by Assange, told The Associated Press that she would "do everything" to have the Swedish case reopened so Assange can be extradited to Sweden and prosecuted.
If found guilty of the US charges, Assange could get five years in prison.
A witness who prosecutors were seeking to interview and an associate of Assange based in Europe - who also requested anonymity - said that before his arrest Assange anxious that USA prosecutors would also bring charges related to WikiLeaks' publication of Central Intelligence Agency computer hacking tools, which the website described as its "Vault 7" cache.
He tweeted: "The extradition of Julian Assange to the U.S. for exposing evidence of atrocities in Iraq and Afghanistan should be opposed by the British government".
"It is therefore of grave concern to us that it appears that the Swedish authorities were not aware of the plans made to arrest Mr Assange yesterday in London, and we would welcome clarity as to what action the United Kingdom authorities took to ensure that the Swedish prosecutors were informed in advance of this decision", it adds.
I've just been with Mr. Assange in the police cells. She said a request by Swedish authorities, however, shouldn't be ignored.
If Britain receives competing extradition requests, lawyers say the Home Secretary would have some leeway in deciding which takes priority.
"After media reports that Julian Assange has been arrested in London, the counsel for the injured party has requested the Swedish preliminary investigation concerning rape be resumed", the Swedish Prosecution Authority said in a statement.
His separate extradition case on charges of computer hacking is set to be next heard by video-link at Westminster Magistrates' Court on May 2. "We've ended the asylum of this spoiled brat", President Lenin Moreno said Thursday explaining his decision to withdraw protection of Assange.
The US official said that within the 60-day period, US authorities could modify or add to the current charges they have filed against Assange.
In a Friday interview with CNN, US Vice President Mike Pence said the United States was "going to bring Julian Assange to justice".
Assange has denied wrongdoing.