He said many of the members weren't that physically able, which would make a relocation difficult.
"Our average age for membership would be 76 or 77 - I can't do what I could five years ago."
He believed it would cost $25,000 - $30,000 to move the shed to other premises.
Hawke said it would likely be the end of Menz Shed if the group was forced to relocate.
"It is so important once you've built something like this up to keep it, it would be devastating for this town to lose it," Hawke said.
Literacy Rotorua manager Mary Longson said she was desperately hoping they would not have to move, because finding another suitable venue would be "very, very difficult".
Longson said the month-to-month lease would be less than ideal for all parties.
"When you start looking to find alternative accommodation, a month isn't much time is it?
"We have started looking [for another premise] but it's just the cost of the tenancy and that you have to take out a three- or five-year agreement, you just don't know where you'll be in five years' time," Longson said.
"Basically if we can stay where we are it would be very, very good."
Longson said she was aware the Menz Shed was going to fight the fact the council would not renew the long-term lease.
"But whether they'll be successful or not, I don't know," she said.
"A lot of people are coming in and saying they think it's appalling that the place might have to close, because it serves a really good purpose in the community - looking after the elderly people.
Council chief financial officer Thomas Colle said in a statement: "Menz Shed has occupied the commercial building, which it sub-leases to Literacy Rotorua and Autism New Zealand, since 2014 under strict conditions - including that the lease could be terminated at any time upon six months' notice.
Colle said the council was "investigating the future of the building".
"Our staff contacted the Menz Shed by phone to suggest an alternative venue but that was declined. Staff are continuing to keep an eye out for any council properties that are becoming available which may be suitable for the Menz Shed.
"No decisions have been made about the future use of the Menz Shed building. But, council plans to inspect the state, use and safety of the building as part of a wider review of all council buildings."