The leadership convention will be held in Moncton on Oct. 27 and the deadline for signing up to vote is next Wednesday.
Party spokesman Rob Dysart says the race is attracting large numbers of voters because the party has made the process more open to the public.
The party has adopted a weighted, one-member-one-vote system that allows voters to cast ballots online, by telephone, via mail-in ballot or at the convention.
As well, there is no cost for voting.
The leadership contenders are Moncton lawyer Brian Gallant, farmer Nick Duivenvoorden and Moncton lawyer Mike Murphy, who served in the cabinet of former premier Shawn Graham.