Boris Johnson is expected to urge US President Joe Biden to increase his country’s climate commitments in a meeting at the White House.
On Monday, the PM had said he was “frustrated” that richer countries were failing to financially support poorer ones fighting climate change.
The two men are also likely to discuss the situation in Afghanistan.
But the PM played down the chances of progress on a UK-US trade deal, calling US negotiators “pretty ruthless”.
Mr Johnson said he would “rather get a deal that works for the UK than get a quick deal”, adding that the president had “a lot of fish to fry”.
The prime minister has travelled to the US for a meeting of the UN’s General Assembly, which is being attended by over 100 world leaders.
Addressing a UN climate meeting on Monday, the prime minister said richer economies had done “nowhere near enough” to support those nations suffering the impact of global warming.
In 2009, developed countries agreed to provide $100bn (£73bn) a year to poorer countries to fight climate change.
By 2019, the figure had reached just over $79bn that year, according to the Organisation for Economic Co-operation and Development (OECD).
The lead author of a UN-commissioned expert report into the pledge, Amar Bhattacharya, says within the G7 [a group of rich countries], Germany, Japan and France have been the leaders in terms of meeting the target.
The UK and Canada are slightly behind them while the United States and Italy are further behind.
Mr Johnson told UN delegates “the gulf between what has been promised, what is actually being delivered, and what needs to happen… remains vast”.
The US’s climate envoy John Kerry raised hopes that there could be progress towards the target, telling Sky News: “I think we’re going to get it done by COP [the climate conference to be held in Glasgow], and the US will do its part.”
Asked if Mr Biden would announce more funds this week, he added: “I’m telling you to stay tuned into the president’s speech, and we’ll see where we are.”
Mr Johnson said it would “send a massively powerful signal” if the US did commit to more funding but insisted “we are not counting our chickens”.
On Monday, the prime minister also met Amazon boss Jeff Bezos. A Downing Street spokesman said they “discussed the upcoming COP26 Summit and agreed there was an urgent need to mobilise more public and private money to help developing countries protect biodiversity”.
The spokesman added that the PM “raised the issue of taxation, and hoped progress could be in implementing the G7 agreement on tax”.
The meeting with the president in the Oval Office will be the first time Mr Biden and Mr Johnson have met face-to-face since a G7 summit held in Cornwall earlier this year.
The prime minister acknowledged that the US-UK relationship had experience some problems in recent years, referring to “pebbles in the shoe”.
But, he pointed to negotiation successes citing the end of the ban on British beef, and suspension of US tariffs on Scotch whisky.
Speaking ahead of the meeting, Mr Johnson described his own relationship with Mr Biden as “genuinely terrific – we see eye to eye on all sorts of things” adding that the two men had bonded over trains.
“He’s a bit of a train nut,” he added.