The prime minister is scheduled to meet leaders of both countries during a 24-hour trip on Wednesday.
Mr Johnson is expected to discuss Europe’s response to the ongoing Russian invasion of Ukraine.
Mr Johnson’s official spokesman said the visit was also about the “security of Europe more broadly”.
“We understand the positions of Sweden and Finland and that is why the prime minister is going to discuss these broader security issues,” he said.
Asked whether the two countries’ possible membership of the alliance would be discussed, the prime minister’s spokesman said: “We support democratic capabilities to decide on things like Nato membership.”
Mr Johnson will give a news conference in each country, travelling to Sweden first before going on to Finland and then returning to the UK.
Nato – the North Atlantic Treaty Organisation – is a 30-nation defensive alliance founded shortly after the end of World War Two.
It has its headquarters in Brussels, but is dominated by the massive military and nuclear missile power of the US.
Support for joining Nato has increased in both both Sweden and Finland since Russia invaded Ukraine, despite their long history of pursuing policies of military neutrality to avoid conflict with regional powers.
Finland and Sweden are both modern, democratic countries that fulfil the criteria for membership.
Nato’s chief, secretary general Jens Stoltenberg, has said the alliance would welcome them with open arms and there would be minimum delay in processing their membership.
During a visit to Sweden in April, Finnish Prime Minister Sanna Marin said “everything had changed” when Russia invaded Ukraine and told reporters Finland must to be “prepared for all kinds of actions from Russia”.
Her comments coincided with the publication of a security report that warned membership of Nato could result in “increased tensions on the border between Finland and Russia”.
At the same time, Ms Marin’s Swedish counterpart Magdalena Andersson told reporters that the same “very serious analysis” was taking place as in Finland and she saw no point in delaying it.
Finland for decades, and Sweden for centuries, have chosen to adopt a kind of neutral status rather than enter any military alliance.
But Russia’s invasion of Ukraine has transformed public opinion in both countries. People seem now to want the protection that Nato membership can provide.
But neither country would get the alliance’s security guarantee – that an attack on one member is an attack on all – until their application has been accepted. That could take some months. Until that point, there is a moment of vulnerability.
So what will be interesting is to see what kind of support the UK – and other countries – might be prepared to provide Sweden and Finland in the meantime, during that so-called “grey zone” between both countries’ application and accession to the Western military alliance.
Finnish Foreign Minister Pekka Haavisto subsequently said it would be “useful” for Sweden and Finland to launch joint Nato membership bids, but added no fixed date had been set for any potential application.
However, Russia has warned them not to and threatened “a military technical response” if they do try to join.
Kremlin spokesman Dmitry Peskov stressed that Moscow would have to “rebalance the situation” with its own measures if any bid went ahead.
In a speech last month, Foreign Secretary Liz Truss said they should be admitted “as soon as possible” if they decided to apply for membership.
Mr Johnson held talks with Ms Andersson and Finnish president Sauli Niinisto in March as part of a meeting of the Joint Expeditionary Force nations, which includes Denmark, Estonia, Iceland, Latvia, Lithuania, the Netherlands and Norway.
After the meeting, No 10 said the leaders had all agreed the invasion of Ukraine had “dramatically changed the landscape of European security”.
Finland shares a land border of 830 miles (1,340km) with Russia and is only about 250 miles from St Petersburg.