The prospect has triggered an urgent appraisal in Western capitals and Israel over handling the political power that the organisation they have long proscribed as "terrorist" is likely to wield in Palestinian politics from the end of this week.
The dilemma is partly over whether Hamas's probable success is a menacing new threat in an already unstable region, or the first stage of a conversion to the ballot box by the Islamic faction which has been responsible for the largest proportion of Israeli deaths in suicide bombings in the past five years.
Hamas's electoral momentum has been evident way beyond its natural stronghold of Gaza. It was tangible for example, from the moment its members and supporters swept to victory in the West Bank city of Nablus on an anti-corruption ticket in the town council elections on 15 December. Everybody who was anybody in the town turned out to congratulate the new mayor, Adli Yaish, a local car-parts dealer. He is prominent in several of the Islamic charities which go a long way to making up for the lack of an effective Palestinian welfare state and on which Hamas has built much of its popular reputation for fair dealing.
True, outside the trade union theatre where his victory reception was held, they were selling DVDs portraying the Hamas "martyrs" to the conflict with Israel to the accompaniment of martial music. But inside, as local community leaders from headmasters to a posse of Christian clergy queued up to embrace Mr Yaish, the green campaign buttons and banners had not a vestige of this. Party organisers handed out baklava to those arriving, as Yasser Mansour, the number seven candidate on the faction's national list, expertly worked the room to exploit the victory.
"In the name of God, the results are not strange," Mr Mansour told visitors. "We can attribute them to two things - corruption [in the Palestinian Authority] and the security chaos that the Palestinian people have been suffering for 10 years."
To the untutored, it may seem odd to hear Hamas condemning "security chaos". But nowhere is the deeply unpopular internal violence to which Mr Mansour was referring more evident than here in Gaza. The Fatah-dominated PA has failed to prevent kidnappings, lethal tribal feuds, and political gun battles - mainly within Fatah itself.
Yesterday, in Gaza City's Shifa Hospital, Suleiman Ashabia winced with pain as he explained how he took seven bullets in his legs and hand on Friday evening. Mr Ashabia, 22, the local deputy campaign manager for the independent Third Way party is "90 per cent sure" the masked gunman who shot him at close range with an AK-47 was political. "We are doing a great job in Maghazi. Whoever it was wanted to stop the hard work. After what happened, people in the Third Way will be afraid."
Posing as a party supporter, one of the gang who targeted him had lured him to a bogus meeting in the market of the Maghazi refugee camp and used a mobile phone call to check what he was wearing. He opened fire as Mr Ashabia returned to his home and then continued shooting him on the ground after he was felled by the first bullet.
The faction least likely to have attacked him, on the face of it, is Hamas. The polls suggest it has next to nothing to fear from Third Way. But it has also fought this campaign with a professionalism and discipline that has eluded most other parties.
You could see it in the horn-honking, banner-waving, 30-vehicle loudspeaker motorcade that drove through the streets of Gaza City yesterday, and in the women's rally which Hamas slickly organised in Nablus last week. That was opened by Hamas's distant equivalent of drum majorettes, with long-skirted, white-scarfed, banner-bearing young women processing into Balur hall to chants of: "The Koran is [Hamas's] constitution and Jihad for the sake of God is its way."
The most prominent local female candidate, Muna Mansour, did not seek to disguise her desire to see sharia embodied in Palestinian law - something feared by secular Palestinians. In talks with reporters she even told - approvingly - a chilling story about the Prophet Mohamed authorising the stoning to death of a self-confessed unmarried mother, though only after the woman's child had grown up.
But Mrs Mansour also presented a more pragmatic side, declaring several times: "Hamas is not the Taliban." Distancing herself from Hamas's reputation for fierce social conservatism, she insisted she was in favour of shelters for battered wives, for an end to "honour killing" and of better education for women. And she suggested that a long-term truce with Israel was possible while believing that an interim phase of two states side by side would not deflect Hamas from its goal of "liberating Palestine from the river to the sea".
This goes to the heart of Israel's and the West's dilemma. Awakening to the new reality - including perhaps the realisation that Israel's effort to disrupt Hamas's campaign by arresting candidates has, if anything, strengthened it politically - some senior figures, from President Moshe Katsav to Shimon Peres, have hinted that Israel could talk to Hamas if it abandoned its militancy and ended its commitment to destroy Israel. Difficult as it is, the first condition may be easier to fulfil than the second.
Skilfully Hamas, having largely adhered to the truce it made last year with Mahmoud Abbas, the Palestinian President, has given little away on post-election plans. But Mr Abbas has always believed that Hamas's assumption of influence will lead it to abandon weapons against Israel and has warned it will not join any coalition with Fatah unless it does so. But Hamas has so far made it clear it has no intention of recognising Israel, as the PLO eventually did under Yasser Arafat.
Mohammed Abu Teir, Hamas's number two candidate, declined to comment on talks though he acknowledged that Hamas councils like Qalqilya's had had "no problem" discussing "technical matters" with Israel. But his message to Israel had been: "You have negotiated with the PLO for 30 years - and what you have given to the PLO?"
On Hamas's notable decision not to include its long-held commitment to the destruction of Israel in its election manifesto, Sheikh Abu Teir said guardedly: "We know how to conduct politics - we are passing through a new stage of politics and we are part of it. We raise the appropriate slogans and there is no need to raise an inappropriate slogan [destroying Israel]. We have a programme that deals with internal issues."
Interpreting that enigmatic statement will be only one of many questions exercising the world's capitals if Hamas fulfils poll predictions on Wednesday.
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