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He will urge international partners to match the UK’s commitments on aid and resettling those most in need.
But UK defence minister Ben Wallace said on Tuesday it was unlikely the US will delay the evacuation of their troops from Afghanistan, as the August 31 withdrawal deadline draws closer.
He said as that date approaches “it gets more and more dangerous".
While an extension would allow more people to be evacuated from Afghanistan, the Taliban has made it clear a delay would be a red line.
“I think it is unlikely. Not only because of what the Taliban has said but if you look at the public statements of President Biden I think it is unlikely,” Mr Wallace told Sky News.
“It is definitely worth us all trying and we will.”
But he hit back at those who said the UK could do more to secure Kabul's airport and therefore evacuate more people.
“It’s not about effectively whether I could fly in thousands of troops and secure the airport.
“Yes I could do that, I could probably secure the airport for a few months, or maybe a year or two,” he told the BBC.
“But for what purpose? For them to be shot at, attacked, people not to get to the airport and to trigger just a permanent fight? I don’t think that is a solution.”
France will also end its evacuations later this week if the US sticks to its current plan, a foreign ministry official said. Almost 2,000 French nationals and Afghans have been rescued and flown to France via a military base in Abu Dhabi.
Spain said it would not be able to rescue all Afghans who served Spanish missions.
“It is a very frustrating situation for everyone, because even those who reach Kabul, access to the airport is very complicated,” defence minister Margarita Robles said.
G7 leaders are expected to reaffirm their commitment to protect the gains made in Afghanistan over the past 20 years, in particular on girls’ education and the rights of women and minorities.
Their discussions will cover a collaboration on evacuations at Kabul airport and longer-term work to secure a more stable future for Afghanistan, as well as ensuring any new government is inclusive and abides by its international obligations.
“Our first priority is to complete the evacuation of our citizens and those Afghans who have assisted our efforts over the past 20 years,” Mr Johnson said.
“But as we look ahead to the next phase, it’s vital we come together as an international community and agree a joint approach for the longer term.
“That’s why I’ve called an emergency meeting of the G7, to coordinate our response to the immediate crisis, to reaffirm our commitment to the Afghan people, and to ask our international partners to match the UK’s commitments to support those in need.
“Together with our partners and allies, we will continue to use every humanitarian and diplomatic lever to safeguard human rights and protect the gains made over the last two decades.
“The Taliban will be judged by their deeds and not their words.”
This week, Mr Johnson set out a five-point plan to address the risk of a humanitarian crisis in Afghanistan.
The plan includes immediately helping those to whom the UK has direct obligations and protecting the country against any threat from terrorism.
Mr Johnson has also placed importance on supporting Afghan people through humanitarian and development assistance, and creating safe and legal routes to resettle Afghans in need.
The plan also includes developing a clear plan for dealing with the new Afghan regime in a unified and concerted way.
Mr Johnson also chaired a meeting of his emergency national security committee on Monday afternoon where ministers discussed the latest situation.
The UK has secured the evacuation of around 8,600 people out of Kabul in the past 10 days, which includes British nationals and their dependants, embassy staff, and Afghan nationals under the Relocation and Assistance Policy programme.
Last week, the British government doubled the amount of humanitarian aid to the region, committing up to £286 million ($392.2m) with immediate effect.
It also announced a new resettlement scheme that will relocate up to 20,000 vulnerable Afghans.