The Australian Financial Review, quoting security sources, said that the country's security agencies had recommended against permitting the company bid for 5G contracts.
If Huawei is banned, this would be the second time the company, which has a presence in 170 countries, has been kept out of major projects in Australia. It was banned from bidding for contracts for Australia's national broadband network, the NBN, in 2012.
Australia also acted to prevent Huawei participating in an undersea cable project linking the Solomon Islands and Papua New Guinea to Sydney.
Contacted for comment, Jeremy Mitchell, corporate affairs director for Huawei Australia, told iTWire: "Huawei remains in productive discussions with the government. It has not been informed of any decision to exclude it from 5G networks.
"In a vast country like Australia telecommunications is vital. Consumers and businesses want fast, affordable and reliable services. Huawei helps to provide that service."
American fears about the company extend back nearly a decade with the NSA having hacked into Huawei's offices in Shenzhen in 2010, according to documents released by whistleblower Edward Snowden.
A report on this hack quoted an NSA document as saying: “Many of our targets communicate over Huawei-produced products. We want to make sure that we know how to exploit these products", to "gain access to networks of interest” around the world.
As iTWire reported on Wednesday, Solomons Prime Minister Rick Houenipwela and Turnbull are set to ink a deal on the first stage of the Solomons project which will cost the Australian taxpayer in excess of $100 million.
The AFR quoted one source, who was claimed to be familiar with the deliberations around Huawei, as saying: "It's hard to see how compromising your telecommunications network is in the national interest. The security agencies found it hard to see that there's any way around it."
Last week, as iTWire reported, during an interview on the ABC's Radio National, Huawei Australia chairman John Lord refused to speculate on what the company would do if it were refused a role in the 5G rollout, though he said it would be a huge blow.
"The future is all supposition and we certainly hope that is not the case, but as I said it's about 70% of our business, we have about 750 Australians employed," he said. "It's just huge and this is still the main part of our business in Australia: mobile broadband."
Huawei supplies equipment to Optus and Vodafone and is expected to also supply TPG which is in the process of launching a mobile network. Telstra gets equipment from Swedish giant Ericsson.