Trump said earlier this week the Navy hospital ships Mercy and Comfort would be pressed into service, one on each coast, as healthcare systems become badly strained during the pandemic.
But US defense officials have said the ships were not yet ready to sail on the emergency mission, with the Comfort undergoing maintenance in Virginia.
CNN reported on Thursday, citing an unnamed US defense official, that the Mercy would be sent from San Diego to the Seattle area, site of one of the nation's worst outbreaks of the respiratory illness that has killed 187 people nationwide and infected more than 11,000 others.
"I respectfully request you immediately deploy the USNS Mercy Hospital Ship to be stationed at the port of Los Angeles through September 1, 2020, to help decompress our current health care delivery system in the Los Angeles region in response to the COVID-19 outbreak," Newsom said in a letter to Trump.
He said Los Angeles, as the nation's second-largest city, would likely be "disproportionately impacted" by the pandemic in the coming weeks.
Trump spoke with several state governors by phone on Thursday, saying the federal government would be helping out U.S. auto companies and might consider a relief package for the hospitality industry as well. It was not clear if Newsom, a Democrat in his first term as California's governor, was on that call.
In a separate letter sent to US congressional leaders, the California governor asked for additional federal funding for unemployment insurance and social safety net programs and aid to small businesses, schools and universities.
US Senate Majority Leader Mitch McConnell on Thursday introduced a $1 trillion-plus package of legislation to provide direct financial help for Americans, relief for small businesses and their employees, steps to stabilize the economy, and support for healthcare professionals and coronavirus patients
Newsom said Wednesday that modeling has shown more than 60,000 homeless people could become ill with the coronavirus in California over the next eight weeks, overwhelming hospitals.