US-based Christian refugee agencies are pleading with Joe Biden to extend evacuations beyond the 31 August deadline.
With the evacuation moving into its final phase, the window is closing to get thousands of vulnerable Afghans out of the country.
World Relief said it was "deeply concerned" about US citizens and vulnerable Afghans "who are at risk of being left behind".
"Many Afghans put their lives on the line in support of our US mission over the twenty-year conflict only to be turned away at the Kabul airport while trying to be evacuated from Afghanistan in recent weeks," it said.
The non-profit is urging Biden to keep evacuating people from Afghanistan "no matter how long it takes".
According to US media reports, some US troops have already been withdrawn from the country as it winds down operations there.
Jenny Yang, senior vice president of advocacy and policy at World Relief, said the US had an obligation to help those at risk.
"The reality is that thousands of women, children and innocent civilians, including some with U.S. citizenship as well as many allies with pending Special Immigrant Visa applications, will be left behind if we do not act immediately to secure their evacuation," she said.
"World Relief urges the Biden administration to fulfill its promise to the tens of thousands of people who helped the U.S. and risked their families' lives, in addition to their own, to help our country. We also urge these evacuations to include other vulnerable Afghans, including women leaders, human rights activists, and ethnic and religious minorities.
"How we respond to the crisis in Afghanistan will define our generation. A lasting scar will be etched into our nation's history if the U.S. leaves thousands of Afghans behind to face the ongoing threat of death.
"When it comes to getting Americans out of Afghanistan, President Biden has said that 'speed is safety'; the same sentiment must be applied to our Afghan allies."
Biden has ignored pleas from other nations, including the UK, to extend the evacuation mission.
The President has said the US is "on pace" to complete evacuations by the deadline.
"The sooner we finish the better," he said.
However, the US said last night that efforts would continue to assist departures beyond the end of the military airlift.
Myal Greene, head of World Relief, is asking people to pray for the safe evacuation of all those in need of escape.
"Until everyone is brought to safety, we must keep the evacuation going," he said.
"The Biden administration must do everything in its power to keep its promises to these brave individuals that our nation would offer refuge to those who risked their lives to serve the U.S. missions.
"We have a moral obligation to protect Afghan lives at this critical hour. To abandon them to the Taliban as part of the schedule set by the Taliban is morally unacceptable."
Church World Service has called Biden's deadline "arbitrary and harmful" and echoed calls for evacuations to continue as long as is needed.
It warns that a failure to extend the deadline will leave "tens of thousands at the mercy of the Taliban".
"President Biden is holding to a deadline that is arbitrary and harmful," said Erol Kekic, Senior Vice President for CWS' Immigration and Refugee Program.
"We have a moral obligation to create a safe pathway out of Afghanistan for all those in danger to US territory—for as long as it takes to get the job done.
"With the Taliban now blocking access to the Kabul airport, attacks on women and children escalating, and the targeting of men and women who worked alongside the United States persisting, President Biden must recognize that his adherence to a hard deadline will cost countless lives."
The final stages of the evacuation have been impeded by a terrorist threat, with the US and UK telling people to stay away from the airport.
Britain's Defence Secretary Ben Wallace has suggested people wanting out of the country make a run for the border.
Video footage circulating on Twitter suggests many have already tried to do that, with chaotic scenes at the Spin Boldak border crossing between Afghanistan and Pakistan.
"If they think they can make it to a third country, that may be a better option," Wallace said on Wednesday.
"I recommend that they try and make it to the border ... because it is higher profile going to the airport – that is where the Taliban will be focusing their efforts at the moment."