The custom is believed to date back to the 19th century, during a period when the Meiji dynasty reformed Japan's complicated naming culture - and encouraged both foreigners and Japanese people themselves to write their family name second when writing in English, part of a broader attempt to conform to international standards.
But this system has long been used inconsistently. As far back as 1986, the government-funded Japan Foundation had decided to use the family-name-first format in its English-language publications, and historical works or academic papers often did too.
In his remarks yesterday, Kono referred to a 2000 report by the Education Ministry's National Language Council that had recommended the use of the Japanese format. That report didn't change things at the time, but as the Foreign Minister noted, it is now a new era.
The arrival of a new emperor has resulted in a new era, named "Reiwa" for two characters that symbolise auspiciousness and harmony. Japan is hosting a number of major events at the start of this period, including the G20 summit of world leaders next month, the Rugby World Cup and the 2020 Summer Olympics.
Trump will be the first foreign leader to meet Japan's new Emperor Naruhito. The US leader has formed an unusually close bond to Abe - even referring to him as "Prime Minister Shinzo" in 2017.
It is unclear whether the US Government will conform to Kono's request. It also remains unclear whether the entire Japanese Government is behind the idea.
Last month, Kono told a parliamentary committee on diplomacy and defence that he writes his name in the Japanese order on his English-language business card, and that this issue should be discussed by the government as a whole.
But Japan Sports Agency Commissioner Daichi Suzuki has said that the public should be consulted before the move.
"We should be deciding after spending some more time examining how discussions among the public are," Suzuki said, according to the Mainichi Shimbun.