No 10 Downing Street announced a list of senior figures with whom Ms Truss will hold talks this weekend, before the queen is laid to rest on Monday.
A meeting with US President Joe Biden, however, has been postponed for four days.
Mr Martin arrived at Downing Street for a meeting with Ms Truss shortly after 10am on Sunday, nodding and saying “good morning” to the assembled press as he entered the door to No 10.
Ms Truss had been due to meet Mr Biden but the talks were cancelled, and moved instead to a “full bilateral meeting” on Wednesday when both leaders will be in New York for the UN General Assembly.
She was also meeting Mr Trudeau and Polish President Andrzej Duda.
Ms Truss met Mr Biden while she foreign secretary but Sunday's now-postponed event was to be their first meeting since she succeeded Boris Johnson as prime minister.
The meetings were taking place against a backdrop of disputes over the Northern Ireland Protocol, difficulties working towards a trade deal with the US and the war in Ukraine.
It is also likely that Ms Truss will hold talks with French President Emmanuel Macron at the UN summit, rather than during his visit to the UK for the funeral.
Mr Macron said he would attend after offering his country's condolences in a call to King Charles III.
He posted a message on Twitter that described the “unbreakable” ties between France and the UK and promised to strengthen the relationship between the countries by “following the path” of queen.
A meeting represent an opportunity for Ms Truss to clear up where their relationship stands, after she sparked controversy during her campaign to become Conservative Party leader by saying the “jury's out” over whether Mr Macron was a “friend or foe”.
Ms Truss and European Commission President Ursula von der Leyen are also expected to meet on the sidelines of the UN summit on Wednesday to discuss the dispute between the UK and the EU over the Northern Ireland Protocol.
The British prime minister is not expected to meet Chinese Vice President Wang Qishan, who is set to attend the state funeral on Monday instead of President Xi Jinping.
China's official delegation is expected to be barred from attending the queen's lying in state in Westminster Hall because of Beijing's sanctions imposed on seven MPs and peers.
No 10 has declined to describe the conversations Ms Truss and leaders of allied countries as formal bilateral meetings, portraying them instead as talks in which to offer condolences over the queen's death, during which politics is likely to come up.
Chevening in Kent will be used as one base for meetings rather than the prime minister's country residence of Chequers, which is said to be undergoing routine maintenance work after Mr Johnson stepped down.
As well as her meetings with world leaders on Sunday, Ms Truss will have an audience with the king before attending his reception for visiting heads of state at Buckingham Palace.
Ms Ardern will meet Ms Truss, the king and Prince William, Prince of Wales, before the queen's funeral.
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New Zealand's leader confirmed the meetings but revealed few details after arriving in the UK, saying she would offer her country's condolences to King Charles.
“I'll simply share the sorrow that New Zealand has and pass on our deepest condolences,” she said in London.
“At the end of the day, although this is a period of transition for him, he has also lost his beloved mother. For us and New Zealand, that's first and foremost.”
Ms Ardern said the queen's death and new king would be the “focus of conversation” when she and Ms Truss met.
She said she Ukraine and the UK's free trade agreement with New Zealand were also likely to feature on the agenda.
“I am sure we will also transact our relationship as well, but it is all within the context of the week of mourning that the UK is currently in,” she said.
As a member of the Commonwealth, New Zealand will bring a delegation of 10, including the Maori King, Tuheitia Paki, a Foreign Office source said.
About 500 dignitaries from around the world will arrive in London for the state funeral, in what is set to be one of the biggest logistical and diplomatic events in the UK.
They will join members of the royal family, former UK prime ministers and key figures from public life at Westminster Abbey, which can hold about 2,000 people, at 11am on Monday.