© EPA/NASA WORLDVIEWSatellite imagery shows Cyclone Veronica just off the north-west coast of Australia
Cyclone Veronica lingered about 95km (60 miles) off the nation's north-west coast on Monday, officials said.
Despite no longer being expected to make landfall, the category two system is moving slowly and has prompted warnings for locals to remain indoors.
Cyclone Trevor, a category four, hit the Northern Territory on Saturday.
Heavy rain and large waves whipped up by Cyclone Veronica threaten to cause widespread flooding in Western Australia, according to authorities.
An area stretching from Karratha to Port Hedland was enduring gale-force winds of more than 125km/h, the Bureau of Meteorology said.
Similar conditions affected the Northern Territory on Saturday when Cyclone Trevor made landfall between the remote communities of Numbulwar and Borroloola.
In Western Australia, the strong winds snapped trees and caused minor power outages but there were no immediate reports of injuries.
Many locals had reinforced their homes with sandbags and stocked up on food and water supplies.
One woman who gave birth at a local hospital during the storm named the child Veronica, local network Nine reported.
'Unusual' twin occurrences
Cyclone Veronica was a category four before it was downgraded at the weekend.
"It really is quite unusual for two cyclones to happen at the same time, particularly two very strong systems," meteorologist Steph Bond told the BBC, adding it had happened only twice in Australia's history.
Comment: Unusual storm activity appears to be increasing. Yesterday a rare south Atlantic tropical storm formed off the coast of Brazil, the first for nearly a decade.
She attributed the timing to a climate phenomenon known as the Madden-Julian Oscillation creating "favourable" conditions for summer storms.
Cyclone Trevor also brought down trees and power lines after making landfall as a category four system.
Authorities said it had caused no major injuries or significant damage to infrastructure, following the region's biggest evacuation effort in nearly 50 years.
Thousands of people in remote communities relocated to regional centres ahead of the storm. Some began to return home on Monday.
Comment: Two powerful tropical cyclones loom over northern Australia forcing mass evacuations