The contest has been so tightly contested that the vote was delayed from its original date of late October until later today in Los Angeles after an earlier Board meeting ended in heated disagreement.
Former England second row and current IRB vice chairman Beaumont appears to have secured 12 votes, while his French opponent is out in front with 14.
Only furious late lobbying could now alter the outcome and prevent Lapasset from winning another four-year term in office.
"I am excited like before you kick off," said a still optimistic Beaumont prior to today's vote.
"If you don't have a bit of apprehension, you don't play your best. I think I can make a difference. There are challenges, big challenges to be faced and it is not going to be an easy ride but, by getting together and communicating, we can improve the game.
"And hopefully by people seeing who I am, they will back me as a fair person, open, honest and decent. That's the way I am. I can't change that.
"I have dedicated a lot of my life to rugby because I have been lucky to have been able to play it at the highest level. Once you have done that you have been very fortunate. I want to try and put something back into the game."
Lapasset's role in helping Rugby Sevens secure Olympic status has won him huge support among developing rugby nations and the recent backing of delegates from Asia, the United States and the Caribbean block look like handing him a narrow victory.
The Home Unions, who, like the other founding nations Australia, New Zealand, France and South Africa, have two votes each, are backing Beaumont.
New Zealand, Canada and Oceania are also in Beaumont's corner as he looks to strengthen rugby's position from the top down.