A post on the Counties Manukau Touch Facebook page said games at Mountfort Park and Gallaher Park had been cancelled this evening, as a sign of respect.
"Our thoughts and prayers are with the whānau as well as those at touch last night, many of whom are still shaken today," the post read.
A "Counties Manukau Day" scheduled to be held at the club was also being postponed until the new year.
Last night, Manurewa-Papakura Ward Councillor Daniel Newman described the incident as a "devastating gut-punch".
"The tragic death of a young girl in a rugby scrum machine accident at Mountfort Park has shocked the rugby community in Manurewa, and will trigger an outpouring of grief and support for her family," he said.
Kia Ora Touch Whanau
The Counties Manukau Day scheduled for this Sunday 16th December has now been postponed until the New Year.
Blessings and Strength to you all. Arohamai te whanau
Posted by Counties Manukau Touch on Thursday, 13 December 2018
"I have spoken at length with the leadership at the Manurewa Rugby Football Club and know Mountfort Park well.
"We are all devastated at what has occurred. Club members are co-operating with the police to assist their inquiry."
Victim Support was assisting the family of the deceased, who were present at the time of the incident.
Police extended their "heartfelt condolences" to the family.
Newman said the specific details around the incident were not known, but he understood the park was being used as part of a touch module.
"A touch module is run at Mountfort Park as part of a programme to encourage active sport. I understand the park was being used for that purpose at the time the accident took place."
Manurewa Rugby Club president Kere Maihi declined to speak to the Herald. He is also the president for Counties Manukau Community Rugby.
In 2009, former All Black hooker Kevin Mealamu's niece died after she was crushed by a scrum machine.
Speaking to the Herald on Sunday after that incident, Mealamu revealed the cushioned pads of the scrum machine had been removed from the steel-framed machine and it had been tipped up on an angle.
"I think some of the young teenage boys might have left the sports equipment standing up, which it shouldn't have been," he said at the time.
"I think she might have fallen under it, but there might have been some kids playing on the other side of it and maybe pushed it down."