The legislation would see the election of nominees every four years in conjunction with municipal elections.
The list of elected nominees would then be given to the prime minister, who would still make the final decision on who becomes a senator.
New Brunswick has 10 senators but there will be three vacancies before the next municipal elections in 2016.
Premier David Alward said those may have to be filled by appointment.
"Certainly, I would support the prime minister if he chooses to fill them. I do believe it is important that seats be filled in a timely fashion," Alward said.
Alward said his preference is that any Senate vacancies be filled during municipal elections when they wouldn't add any extra financial burden to the province.
But interim Liberal leader Victor Boudreau says Senate elections are not an issue New Brunswickers are concerned with.
"I think it's a total waste of time. It's certainly not a priority," Boudreau said.
"This is simply David Alward bowing to the wishes of Stephen Harper to be one of the first provinces out with this, so he can say 'Look, look, I did what you asked of me.' "
Alward said he is a supporter of the prime minister's efforts to reform the Senate.
"The Senate, in its form now, is not as effective as it could be. We believe that it is important to have that sober second thought," Alward said.
Boudreau said he also supports Senate reform, but it needs to be done with the consensus of all the provinces.
The New Brunswick bill recommends that the Electoral Boundaries and Representation Commission divide the province into five senatorial districts, with two senators in each.
The bill is expected to be referred to the Law Amendments committee for further study and return to the legislature during the fall session.
In a news release Friday the federal government welcomed New Brunswick's move.
"The Harper government applauds New Brunswick's initiative towards a more effective and democratic Canadian Senate," said Tim Uppal, minister of state for democratic reform.
"Senators introduce, review and pass laws that affect Canadians everyday, so it is only right that Canadians have a say in who represents them in the Senate."
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 federal government's Senate reform bill, which would require senators to retire after serving a non-renewable nine-year term and provide provinces with the opportunity to create a process for electing Senate nominees.