Traditionally, the job has been done by the person who runs Conservative Central Office, but Tory MPs who fear for their seats want "Hezza" to take charge.
A Central Office insider said: "It is a serious question. It has to be resolved, and it hasn't been. Nobody has the answer yet. It is very difficult to call. I have never had any doubt it would be Mawhinney - but certainly there are people who take the view that it should be Heseltine."
Dr Mawhinney, an abrasive Ulsterman who is close to John Major, is admired for his cutting attacks on Labour, but Mr Heseltine, many Tories say, has a more "positive" image.
A Tory backbench supporter of Mr Heseltine said Dr Mawhinney's hard- nosed style "puts people off" and in the difficult election coming up, his rival for the job would cut a more persuasive figure.
The party chairman, who tried but failed to halt last night's screening of the BBC soap Casualty, on the grounds that it was "a vehicle for inaccurate political propaganda", took over at Smith Square four months ago from gaffe-prone Jeremy Hanley. Last week he lost his head of communications, Hugh Colver.
The Central Office source said: "It needs to be resolved sooner rather than later. That needs to be clear from a good way out, so that everything beds down and works properly. You don't want that happening too close to an election and causing ructions.
"Ultimately, the Prime Minister will have to decide. If he came down on the side of using Heseltine as No 1, he would have to carry Mawhinney with him to make sure he doesn't throw his toys out of the pram."
Mr Heseltine will speak at the CBI conference in Birmingham tomorrow, before going into hospital on Tuesday for an operation to remove a kidney stone. He will miss the state opening of Parliament and the Queen's Speech this week.
Tony Blair will also address the CBI conference tomorrow, telling them that Labour's changes are "real and irreversible", and the party's new Clause IV commits Labour explicitly to the market "but the market cannot do everything".
He will add: "The role of government in a modern economy is important, but it is also limited: providing a secure, low-inflation environment and promoting long-term investment, raising standards of education and training, building a properly-functioning, first-class infrastructure, seeking to open markets for our goods round the world and creating a strong and cohesive society in which the aim is to provide each citizen with a stake."
Labour's objectives, he will insist, "can be fulfilled in partnership between industry and government".
Lord Tebbit, a former party chairman, who was known as the "Chingford Skinhead" for his bruising style, yesterday appealed to politicians not to be so nasty to each other.
"I was always a pretty tough guy," he said, "but I never hesitated to say to my opponents kind things and decent things to emphasise that they were not all bad."
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