Pressure is mounting on beleaguered Culture Secretary Jeremy Hunt, as a Westminster sleaze watchdog called for a swift investigation of allegations over his handling of News Corporation's takeover bid for BSkyB.

The call from the chairman of the Committee on Standards in Public Life, Sir Christopher Kelly, came as Mr Hunt announced he was handing over text messages and emails relating to the bid to the Leveson Inquiry into media standards.

Prime Minister David Cameron has resisted calls to refer Mr Hunt to his independent adviser on ministerial conduct, Sir Alex Allan, over claims he secretly favoured the bid by Rupert Murdoch's company, arguing that the issue should be left to Lord Justice Leveson.

Deputy Prime Minister Nick Clegg called for Mr Hunt to appear before the inquiry "as quickly as possible" to give his version of events under oath.

And Lord Justice Leveson himself said on Wednesday: "Although I have seen requests for other inquiries and other investigations, it seems to me that the better course is to allow this inquiry to proceed."

But the CSPL chairman said Sir Alex was the "obvious" person to conduct a probe. And he said that if the issue was to be left to Leveson, Downing Street must make clear that the judge has the power to look into ministerial conduct - something which Mr Cameron's official spokesman has repeatedly said is a matter for the Prime Minister.

Sir Christopher said there was "no doubt" that the allegations needed to be "properly investigated".

He added: "If it is to be done instead by Lord Leveson as part of his inquiry then it needs to be clear that all the standards issues, including those relating to the ministerial code, are regarded as being within his remit and will indeed be looked at. It would be helpful to have that put beyond doubt."

Mr Hunt said he was "confident" that the release of his emails and texts would show he handled the BSkyB merger process with "total integrity".

Aides confirmed that the submission would include not only messages sent to Mr Hunt's special adviser Adam Smith, who resigned on Wednesday after the exposure of his contacts with a News Corp lobbyist, but also other communications within his Department for Culture, Media and Sport.