The Al Noor mosque was one of two targeted in the March 15 terror attacks that claimed the lives of 51 people.
"This type of imagery has no place in Aotearoa New Zealand," he said.
"It is abhorrent and will not be tolerated."
Police have increased patrols around Al Noor and Linwood mosques and will be maintaining a visible presence as the anniversary of the terror attacks nears.
"On behalf of NZ Police, I want to thank those members of the community who alerted us to the recent threat against Al Noor mosque," Price said.
"We ask that you continue to remain vigilant, and encourage you to report anything suspicious or concerning to police immediately."