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A spokesman for Palmer said the documents were due to be filed by close of business May 25.
WA closed its border to the rest of Australia on April 5 to combat the spread of COVID-19.
Palmer, who was denied entry to WA last week, declared his banning was unconstitutional.
"There is no time to waste in getting people back to work and building prosperity for the state," he said.
"Closing down the border is not only an act of stupidity by WA premier McGowan, but it is against the Australian constitution and I'm confident the High Court will see it that way as well."
WA premier Mark McGowan has refused to back down on the border closure saying it could be in place for "many months" and would likely be the last restriction to be lifted.
Indeed, the state still has several intrastate borders in place restricting travel between regions as well as the hard border closing the state off from the rest of the country.
Palmer said McGowan had to act now to ensure a successful future beyond COVID-19 by opening his borders to allow trade and travel from Australian states into WA.
"It is a matter of national importance that the borders be open," he said.
WA has six active cases of COVID-19 remaining.
Palmer also criticised McGowan yesterday over royalty discounts offered to CITIC Pacific Mining at its Sino Iron project in the Pilbara.
Palmer's company Mineralogy owns the deposits CITIC Pacific is mining.
"Serious questions must be asked as to why Mark McGowan and his government have such a close relationship with China," Palmer said.