During testimony from ex Sky chairman James Murdoch, the inquiry discussed a series of News Corp emails that appeared to suggest the minister had privately expressed his backing for the bid.
Following the revelations, Labour deputy leader Harriet Harman told MPs that Hunt's conduct during the bid had fallen "woefully short" of the standards expected.
However, Prime Minister David Cameron has stated his "full confidence" in Hunt.
A spokesman for the prime minister also insisted that Cameron had "done nothing wrong", after Murdoch confirmed that he had also discussed the Sky bid with him in late December 2010.
News Corporation withdrew its bid to acquire the remaining shares in pay-TV giant Sky that it did not own in July 2011, following the phone hacking scandal at the News of the World.
Hunt had taken over arbitration of the takeover in December 2010 after the power was stripped from business secretary Vince Cable, who had been secretly recorded saying he had "declared war" on Rupert Murdoch's media empire.
Fresh questions have been raised today over Hunt's stance on the takeover after News Corp submitted its first bid for Sky in June 2010.
This follows the release of a series of emails from Frederic Michel, the head of public affairs at News Corp.
Another email, dated from January 2011, said: "JH believes we are in a good place tonight."
Speaking at the inquiry, Robert Jay QC suggested that Hunt was liaising via Michael as a way to avoid any suggestions of inappropriate contact over the proposed takeover.
However, Murdoch said that it was "acceptable and part of the process". He also denied that Hunt had acted as "cheerleader" for News Corp's bid.
Michel's written statement to the inquiry also claimed that he had never had direct contact with Hunt, despite the emails suggesting that this was the case.
BBC News reports that Hunt will resist calls to resign and instead give evidence in person at the Leveson Inquiry into press ethics and standards.
> James Murdoch stands by hacking testimony at Leveson Inquiry