Israel and Switzerland both said they identified one infected person who had recently travelled abroad. Israel is investigating other suspected cases.
Monkeypox does not tend to spread easily between people and the illness is usually mild.
The virus is most common in remote parts of Central and West Africa.
More than 80 cases have been confirmed in the recent outbreak in Europe, the US, Canada and Australia.
This outbreak has taken scientists by surprise, but the risk to the wider public is said to be low. Most people who catch the virus recover within a few weeks, according to the UK's National Health Service.
The World Health Organization has said a number of other suspected cases are being investigated - without naming the countries involved - and warned that more infections are likely to be confirmed.
Asked about the outbreak as he finished a visit to South Korea, US President Joe Biden said that if the virus were to spread more widely it would be "consequential", adding that "it is something that everybody should be concerned about".
He said the US was "working hard" on its response and what vaccines in might use.
After the outbreak was first identified in the UK, the virus began to be detected across Europe - with public health agencies in Spain, Portugal, Germany, Belgium, France, the Netherlands, Italy and Sweden all confirming cases.
The UK Health Security Agency has identified 20 cases so far and its chief medical adviser Dr Susan Hopkins told the BBC's Sunday Morning programme: "We are detecting more cases on a daily basis."
She said the virus is now spreading in the community - with cases detected which have no contact with anyone who has visited West Africa, where the disease is endemic.
But the risk to the general population remains "extremely low", with cases so far mostly found in some urban areas and among gay or bisexual men, Dr Hopkins said.
Although there is no specific vaccine for monkeypox, several countries have said they are stocking smallpox vaccines, which are about 85% effective in preventing infection because the two viruses are quite similar.