Bourque was sentenced last month to life in prison with no eligibility for parole for 75 years after pleading guilty to three counts of first-degree murder and two counts of attempted murder. The penalty is the harshest Canada has seen since the last state-sanctioned executions in 1962.
A number of news media organizations including The Canadian Press have asked for access to the exhibits that were used at his sentencing hearing.
Lawyers for Bourque and the Crown are opposing the request.
David Coles, the lawyer representing the media, said the public has a right to see and hear the evidence.
"It seems to me the Canadian people have a right to review the evidence upon which the chief justice made the decision that was appropriate," Coles said.
"We live in a society where unfortunately over the past several years a multiplicity of our peace officers have been gunned down and one has to ask, 'Are the protocols appropriate? Are their orders appropriate? Is their equipment sufficient?'"
Crown prosecutor Cameron Gunn declined comment in advance of Monday's hearing. Bourque's lawyer, David Lutz, did not return a request for comment.
Coles said the evidence should also be made public because while it was presented in open court, not everyone could be there to hear it, and the information it conveys is of interest throughout Canada.
"This case where you have this kind of ambush and premeditated gunning down of RCMP members is a public interest in Winnipeg, just like it's a public interest ... in Moncton," Coles said.
"Is the evidence sought to be kept from the Canadian public of a nature that it warrants an extraordinary order to seal it from examination by people?"
The exhibits include recordings of 911 calls, police radio transmissions from the five officers shot and an interview Bourque gave to police after his arrest June 6.
During that interview that lasted about three hours, Bourque said he used a semi-automatic rifle that he legally obtained to shoot the five officers in Moncton's north end. Bourque said he then fled into the woods near a suburban neighbourhood, where he was arrested 28 hours later.
He rambled on at length about a number of issues including his Catholic upbringing, climate change, class warfare and his anger toward police, whom he described as "soldiers" of the state.
Bourque killed constables Dave Ross, 32, Fabrice Gevaudan, 45, and Doug Larche, 40.
Constables Eric Dubois and Darlene Goguen were also injured in the shootings, which brought the city to a standstill for nearly 30 hours before Bourque was arrested.
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