The leaders are expected to reaffirm a vow to safeguard gains made in Afghanistan over the past 20 years, particularly around girls' education and women's rights, Johnson's office said.
The meeting will also discuss possible economic sanctions and whether to withhold aid if the Taliban committed human rights abuses or allowed their territory to be used as a haven for militants, British sources said on Sunday.
There will also be a call for unity over when and whether to officially recognise the Taliban as the leaders of the country, diplomatic sources said.
But it is the evacuation effort that is likely to dominate discussions. The Taliban have said the Aug 31 deadline is a red line, but Biden has said troops might stay beyond that date.
Johnson spoke to Biden ahead of the G7 meeting, when they agreed to ensure that all those eligible to leave Kabul were able to do so "including after the initial phase of the evacuation has ended".
The president's handling of the crisis so far has angered some of the United States' closest allies.
During an emergency session in parliament last week, British lawmakers, including many senior figures from Johnson's own Conservative Party, lined up to condemn Biden's actions, while former Prime Minister Tony Blair described the politics behind the withdrawal as "imbecilic".
France has said more time was needed for the evacuation and German Foreign Minister Heiko Maas said on Monday the G7 needed to consider whether it should remain beyond the deadline set by the United States.
British Defence Minister Ben Wallace said there was little prospect of British forces staying on to continue an evacuation once US forces had left.
"The prime minister is obviously at G7 going to try and raise the prospect of seeing if the United States will extend," he said. "It's really important for people to understand, the United States have over 6,000 people in Kabul airport and when they withdraw, that will take away the framework that has allowed us to withdraw, and we will have to go as well."