Although Mr Taylor said he would not announce any decision to run for the post until he had consulted widely within the party, Unionist sources said that he probably has the most support among the 850 delegates who will elect the leader.
Mr Maginnis is the party's most accomplished media performer, but suffers the advantage of being very popular with Catholics. Many of those in the party grassroots, who are to vote in the election in Belfast's Ulster Hall on 8 September, regard him with suspicion as being too liberal for their taste.
Another contender, William Ross MP, yesterday set out his stall as the candidate of the status quo, describing himself as: "A safe pair of hands in the constant quest for stability in Northern Ireland - real stability... There would not be any sudden swerves or changes."
Mr Maginnis said he would be prepared to talk with Sinn Fein if it renounced violence.
"We accept that anyone who eschews violence totally cannot be precluded from the democratic system, irrespective of what has happened in the past. But it will be difficult to sit down with these people," he added.
Mr Molyneaux, meanwhile, was entertained at a private farewell lunch at Chequers by John Major and the Secretary of State for Northern Ireland, Sir Patrick Mayhew. This was described as a mainly social occasion arranged to recognise Mr Molyneaux's long parliamentary career and his close working relationship with the Prime Minister. Ironically, many Unionists regard Mr Major as having let down Mr Molyneaux by pursuing the peace process with Sinn Fein and by mapping out, in the framework documents, a political path which they see as designed to weaken rather than strengthen the Union.
Meanwhile, there were indications of a thaw in relations between Dublin and London when the Irish foreign minister, Dick Spring, said he expected a summit involving Mr Major and the Taoiseach, John Bruton, to go ahead on 6 September. Last weekend there was speculation that this might be postponed as a sign of Dublin's displeasure with the slow pace imposed by London on the peace process.
In Belfast yesterday Sinn Fein and the Social Democratic and Labour Party (SDLP) issued a joint statement urgently calling for all-party talks. They asked the British and Irish governments to set a date in September to initiate what they described as "this critical phase of the peace process".
This came after a meeting on Monday between Sinn Fein and Irish ministers, following which a joint statement said that the "clear priority now is a comprehensive and inclusive all-party talks process".
These moves are apparently designed to increase pressure on London to end the deadlock on arms decommissioning and speed progress towards talks. London's position, reiterated only last week, is that decommissioning should begin before talks take place.
Mark Durkan, the SDLP chairman, said yesterday: "I hope that whoever is the new Unionist leader takes this opportunity. I am quite happy that whoever is Unionist leader can make whatever challenges they want to us and other parties in dialogue - the real issue is how do we move forward to dialogue."
The four main contenders to succeed Molyneaux
Aged 59, he is a classic Unionist of the "not an inch" school, a man who is temperamentally suspicious of new ideas and who has never publicly flirted with anything outside the strictest orthodoxies of Unionism. He is therefore the principal candidate of the status quo.
He was once caricatured as a tree after a nationalist politician accused him of "giving his usual impression of a block of wood".
A former member of the paramilitary B Specials, he has represented safe Londonderry seats in the Commons since 1974. A member of the Monday club, he favours the death penalty, opposes devolution and advocates Northern Ireland's integration with Britain.
The Rev Martin Smyth
The 64-year-old Presbyterian minister's chief claim to fame is that he is a member of the Orange Order. Although members of this Order are sometimes involved in controversial marches which can, on occasion, lead to disorder, Mr Smyth has a reputation for keeping away from the scenes of any disturbances.
A long-time right-winger who has opposed accommodation with nationalists, he is a relatively low-key politician who does not court publicity.
Mr Smyth is presenting himself as a safe pair of hands; as someone who would pursue a "steady as she goes" policy.
In religious terms he is strongly anti-ecumenical.
The 57-year-old has done much to modernise his party's image and to take its arguments to the outside world. He has done so by becoming its most polished television performer and by becoming a frequent visitor to the Irish Republic, where his readiness to take part in debate and dialogue has made him the south's favourite Unionist politician.
A major in the Ulster Defence Regiment for many years, Mr Maginnis has combined political liberalism with a hardline security stance. He is a long-time advocate of internment without trial and has, on several occasions, predicted that the IRA intends to end its ceasefire.
Aged 57, he has been a politician since the Sixties. He has always been regarded as a hardliner, going back to the days when he opposed the reformist policies of the late Captain Terence O'Neill.
MP for Strangford, he served two terms in the European assembly before entering the Commons in 1983. For most of his career he has been a highly individual politician ready to express opinions which were often sharply at odds with the party leadership.
Some nationalists support his candidature in the hope that he would be far-seeing enough to realise that the Unionists should make a deal with the Catholic community.Reuse content