Labour has urged the government not to "abandon" Afghan workers who helped British forces by allowing more of them to settle in the UK.
Many interpreters and other former staff are fearing for their lives as the Taliban continues to extend its control over the whole of Afghanistan.
Shadow home secretary Nick Thomas-Symonds said the UK had to put in "specific and safe" asylum routes.
The government has said it should get more than 1,000 people a day to the UK.
Defence Secretary Ben Wallace told the BBC the biggest barrier to increasing the rate of evacuation was not a lack of capacity on planes but "how quickly" people can be processed by officials.
US forces have secured the perimeter of Kabul airport, where hundreds of Afghan and foreign workers are waiting for flights out.
The UK has also sent around 600 UK troops to assist the withdrawal, with Prime Minister Boris Johnson promising to get "all those who have helped the UK effort over 20 years" out of Afghanistan "as fast as we can".
Labour leader Sir Keir Starmer said all those who had served deserved protection, adding that there should be no "nice distinction" between workers based on their professions or whether they had been UK government employees or contractors.
Mr Thomas-Symonds has written to Home Secretary Priti Patel, asking for an update on the numbers of those being helped.
"The situation in Afghanistan is truly awful," he said. "We must now live up to our obligations, especially to those Afghan people who worked so bravely with British representatives in Afghanistan.
"Our resettlement scheme must urgently be expanded to ensure people to whom we owe a huge debt are not abandoned."
Last week the government said about 2,000 Afghan interpreters and "other people we have an obligation to" would also be transported to the UK, joining about 3,000 who have already been taken out of the country.
Mr Wallace told BBC Breakfast: "Our flights, our planning and coming in and out and soon if we manage to keep it in the way we're planning to, we should have capacity for over 1,000 people a day to exit to the UK.
"Currently this is not about capacity on planes; it's about processing speed. So that's why we're trying to fix that."
Mr Wallace also said the rules had change to allow more contractors who had served British forces into the UK, and that he would speak to the home secretary about further moves to process as many Afghan passport holders as possible.
Downing Street has said plans for wider asylum and any updates to eligibility for the resettlement scheme will be set out in the coming days.
But Mr Wallace added: "There will be some people left behind. We made that clear in the last few weeks. I'm not going to raise expectations."
Ministers would "do our very best" to get those eligible out of Afghanistan by 31 August, he added.
Appearing on LBC radio, Mr Wallace was close to tears as he said that "some people won't get back", adding: "It's sad the West has done what it's done."
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