Space was required for the group's machinery, and with the clubrooms being two-storeyed, it would be difficult for the group's older members to get up and down the stairs, he said.
In a statement the council said it and the Menz Shed agreed in December the group would receive three months' notice before it needed to vacate the premises.
"We don't yet know when that will be as there has been no decision made as yet about the future of the property.
"In the meantime, council staff are keeping an eye out for any council properties that are becoming available and may fit the needs of the Menz Shed. So far we haven't identified anywhere."
Wilson said the group had between 50 and 60 members, who were feeling "not very good" about the situation.
"They're all sort of retired, this is the only sort of place that can help them through, for a lot of reasons.
"It's good for their health, both mentally and physically."
The work the group did for the community was "very important", he said.
With the group limited in the spaces they can set up in, they were worried they would have to shut up shop altogether.