Abe's request to meet Trump before he takes office reflects growing concern over US foreign policy in Japan.
Some of Trump's campaign promises have called into question longstanding agreements between the two countries, such as the presence of US military bases in Japan.
Trump has also vowed to block the Trans-Pacific Partnership (TPP), a vast free trade deal involving Japan, the US and 10 other Pacific Rim countries.
"It's a smart move from the Japanese prime minister," says Barthélémy Courmont, researcher at the French Institute for International and Strategic Affairs.
"We've seen over the past few days that Trump is hesitating and taking positions that are not as tough as what he said during his campaign. So Abe is trying to take advantage of this very particular period of time in between two presidencies in order to influence the newly elected US president about the necessity to keep the TPP."
The alliance between the US and Japan since the end of World War II has been crucial to the east Asian country's defense and economy.
It is significant that Abe is the first foreign leader meeting with Trump, according to Andrew Staples, Director for the Economist Corporate Network in south-east Asia.
"Japan is one of the United States' staunchest allies," he says.
Staples explains that Trump's call to withdraw American troops from Japan "has caused great concern in the region".
"America's commitment to the post-war order in east Asia seems to be questioned, so that's a major issue that Abe will want to get some reassurance on."
US security umbrella
The US has 85 military facilities and more than 50,000 troops in Japan, whose constitution forbids it from using force to settle international disputes.
This clause, imposed on Japan by the US after World War II, limits its armed forces to a strictly defensive role.
"The question around the US security umbrella is basically the most important partnership and strategic alliance across the Asia-Pacific region," says Hosuk Lee-Makiyama, director of the European Centre for International Political Economy.
According to Lee-Makiyama, Trump's questioning of the merits of the US security umbrella in the region amounts to "questioning the existence of Japan as a country".
Trade and the pivot to Asia
During his campaign Trump strongly opposed the Trans-Pacific Partnership, a trade agreement involving the US, Japan and 10 other Pacific Rim countries.
The deal, which has yet to be ratified, was championed by President Barack Obama as part of his "pivot to Asia" strategy.
China, however, is not included in the TPP, which is a big reason why Japan wants it ratified.
"The Japanese really believe in the possibility of the TPP to counter the rise of China," says Courmont.
"They believe that this agreement is the best guarantee for them to remain a significant power in east Asia."