A draft copy of the proposed legislation — obtained by The Canadian Press — says the election of nominees would be every four years in conjunction with municipal elections, but could be held at any time if necessary.
Voters would choose nominees for existing vacancies, or any expected vacancies that would occur within the next four years.
The list of elected nominees would be given to the prime minister, who would still make the final decision on who becomes a senator.
New Brunswick has 10 senators.
The Act recommends that the Electoral Boundaries and Representation Commission divide the province into five senatorial districts, with two senators in each.
Progressive Conservative Premier David Alward first promised the election of Senate nominees last fall during a speech to his party's annual general meeting.
At the time, Alward said he supported Prime Minister Stephen Harper's Senate reform agenda, and would support any term limits in the red chamber set by the prime minister.
Last June, the federal government introduced the Senate Reform Act which would limit senators to a single nine-year term, but it still has not been passed.
New Brunswick's proposed legislation is expected to be referred to the legislature's Law Amendments committee for further study and would likely return to the House during the fall session.
Last month, voters in Alberta chose three senators-in-waiting, and B.C. Premier Christy Clark has promised her province will follow suit.
The Quebec government has formally launched a court reference challenging the constitutionality of the Senate reform bill.