The election for rugby's top administrator was due to have taken place in Auckland ahead of the World Cup final in October, but the meeting broke up in acrimony with a vote deciding the election would be delayed until Monday.
Early indications show Lapasset may just have the edge over Beaumont, the current vice-chairman, but the vote is expected to be tight.
Both men have a great deal on the line, with Beaumont facing being voted off the IRB executive if he loses.
Lapasset, on the other hand, has ambitions of becoming a member of the International Olympic Committee, having been instrumental in rugby's successful bid for a place in the 2016 Rio Games which has led to increased funding for smaller nations.
Frenchman Lapasset, who speaks three languages, has focused his policies on building upon that Olympic effect, by expanding rugby's reach and growing the game in new areas.
In contrast, Beaumont's focus is top down, strengthening the core Tier One unions who generate the majority of the money. He is the quintessential rugby man against the rugby politician.
Beaumont could stand again as vice-chairman but there are already two candidates for that role, in Graham Mourie from New Zealand and Oregon Hoskins from South Africa.
With both candidates for the chairmanship coming from Europe, the vote is not expected to be split along the north-south divide but on policy and which man delivers the best leadership for the IRB.
Beaumont is thought to have strong backing from the home unions and he could also be supported by New Zealand and Australia.
Lapasset's power base is expected to come from France and Italy, but mainly the unions who would benefit more from global expansion.
Much of the IRB's policies for the next decade are already mapped out.
The IRB is already committed to a review of World Cup finances, as called for by New Zealand and Australia during the last tournament, and there is a 10-year tours agreement in place.
The venues for the 2015 and 2019 World Cups have already been decided.