Asbestos Disease Foundation of Australia president Barry Robson attended a meeting held between families affected near asbestos-contaminated telecommunications pits and Telstra officials on Tuesday.
"Clean the house, basically; the carpets, bedding, front lawns, and one back lawn," he told ABC radio on Wednesday of the demands on the telco.
The families, several of which are still in hotels, are feeling "very stressed".
Telstra is expected to give its response to the demands later on Wednesday.
Asbestos was found last week at a Telstra pit in Penrith, with further discoveries at telecommunications works in Ballarat, Perth, Adelaide, and Tasmania.
Telstra has accepted responsibility for remediation works on its pits and ducts.
The telco is remediating its pits so the builder of the National Broadband Network (NBN) can roll out its fibre-optic cable network.
Four contractors who have been working on the pits in Penrith are undergoing medical tests for asbestos exposure.
Unions are demanding work on the NBN rollout cease until Telstra and NBN Co meet demands on workplace safety.
Communications Electrical Plumbing Union official Dave Mier said his union wants a "fair dinkum" audit before any further work is done.
"We don't want any work to progress on the network unless the people doing the work are suitably qualified," he told ABC radio.
Mier said the process of assessing the pits in Tasmania has been an "absolute joke".
"There have been no audits at all."
Independent Senator Nick Xenophon renewed his call for an investigation by the auditor general.
"When you have people who are virtually refugees from their own homes, having to move out because of this fiasco, then you know there has been a systemic failure of governance and of risk management."
Government backbencher MP Richard Marles said Labor had a good record on dealing with asbestos issues.
"We have got runs on board," he told Sky News. "If you look at the Howard government's record, there is nothing on the board."
Liberal frontbencher Jamie Briggs said it was sad Labor was playing politics on asbestos.
"It just shows the chaos and dysfunction in this place with a government in a desperate desire to find a scare campaign," he said.
NBN plans will unearth more asbestos
The federal opposition's communications spokesman said the company behind Australia's NBN will next year double its contracted workforce, and potentially unearth hazardous asbestos at many more sites.
Malcolm Turnbull warned on Tuesday that the amount of asbestos disturbed by NBN Co construction work on Telstra-managed telecommunications pits and ducts would rise under the government-owned builder's plans for 2014.
Turnbull also accused the Labor government of stirring up "needless anxiety" over concerns that NBN work so far may have exposed people in Western Sydney, Ballarat, Perth, Adelaide, Tasmania, and Queensland to the asbestos fibres that cause mesothelioma.
He said that NBN Co plans to double its contractor workforce to 15,000 next year.
"At the same time as they're going to be heavily audited over asbestos, at the same time several of the contractors have called a stop-work [meeting] over asbestos, they're going to double the size of their contractor workforce and seek to make them 10 times as productive," he told the ABC.
"This is a very big, complex project with much more disturbance."
He also accused Workplace Minister Bill Shorten of "trying to turn this into a national panic".
"In most cases, as long as it's not interfered with, as long as it's not broken up ... then there isn't a health risk," Turnbull said.
On Monday, the government announced an independent task force to monitor work by Telstra to prevent asbestos exposure, including training and supervision of contractors.