Mr Zuckerberg (pictured below) appeared before the US Congress in April but has repeatedly refused to appear in London.
Allan appeared after the committee's chairman, Damian Collins, took the unusual move of seizing a trove of confidential internal Facebook documents from a visiting US tech executive. "Certainly within the next week or so, that would be my hope".
The documents contain revelations Facebook has been fighting to keep out of the public domain, The Observer newspaper reported.
Collins also said that he is urging Facebook to stop blocking the documents from being unsealed in a California court.
After the hearing, the representatives from nine parliaments signed a declaration calling for global rules to govern the internet.
The Digital, Culture, Media and Sport Select Committee, which held the hearing, tweeted, "9 countries". An empty seat was laid out for the CEO.
"While we were playing on our phones and apps, our democratic institutions - our form of civil conversation - seem to have been upended by fratboy billionaires from California", said Mr Zimmer's vice-chairman, Charlie Angus.
Among the concerns raised by lawmakers was Facebook's policies regarding third-party application developers and the use and collection of user data.
Collins wrote in a response that he had the power to publish the documents, which were seized from the owner of the app developer while he was visiting London. In addition, Definers Public Affairs pitched negative stories on other tech giants, Google and Apple, highlighting their policy issues and downplaying Facebook's shortcomings. He was told he risked fines and even imprisonment if he didn't hand over the documents. "And what we learned in 2014 you became aware they were highly inflated and did nothing".
Wedding festivities begin for Priyanka Chopra, Nick Jonas with puja
Nick , 26, posted an adorable photo holding his wife-to-be, 36-year-old Chopra, along with the group on his Instagram story . Priyanka's brother Siddharth Chopra also accompanied the soon-to-be bride and groom for the dinner date.
The Liberal Democrat peer and Facebook executive faced stern questioning on Tuesday over a range of criticisms including recent data breaches, allegations of business malpractice, repeated electoral interference and the rampant spread of disinformation and hate speech on the platform. He said the document indicates a Facebook engineer notified his superiors in October, 2014, that "entities with Russian IP addresses" were pulling more than three billion data points a day from Facebook.
Allan said that information was "at best partial and at worst potentially misleading".
A spokesperson for Facebook said that "the engineers who had flagged these initial concerns subsequently looked into this further and found no evidence of specific Russian activity".
Later on Tuesday, Facebook released a copy of the email chain with the engineer's warning.
He also appeared unable to placate committee members' anger at Zuckerberg's refusal to show up.
"What we're regulating. are the symptoms", said Angus, vice chairman of the House of Commons' standing committee on access to information, privacy and ethics.
The committee asked a series of hard questions, focusing on Facebook's efforts and overall inability to curb the myriad problems plaguing the platform. Asked how Zuckerberg's absence might appear, though, Allan later said it was "not great".
"If you say to me was the election won because of this mass of (campaign spending) activity here, or because of this small amount of activity over here, I think it's reasonable to say let's start by looking - if we think there is a problem - at campaign spending generally".