With the UK still hoping to evacuate thousands more people, the Prime Minister is set to urge the US president to delay the withdrawal of forces from Kabul airport during a summit of G7 leaders.
However, the Taliban said any attempt to continue the operation past August 31 would “provoke a reaction”.
About 200 soldiers from 3 Scots the Black Watch have now left their base in the Highlands in readiness to to be deployed to the country.
Mr Wallace, who visited Fort George to speak to the soldiers ahead of the potential move, said America’s exit will mean “we will have to go as well”.
Ahead of the talks on Tuesday, he said: “The Prime Minister is, obviously at the G7, going to try and raise the prospect of seeing if the United States will extend.
“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… and we will have to go as well.
“I don’t think there is any likelihood of staying on after the United States. If their timetable extends even by a day or two that will give us a day or two more to evacuate people.
“Because we are really down to hours now, not weeks, and we have to make sure we exploit every minute to get people out.”
Mr Wallace said “real problems” were being created by “crowds of many people, who don’t meet any criteria but who are trying to get out of the country”.
And he added: “If we get more days, we get more people out, if the flow improves at the gates.”
He said: “The hardest thing is, to be honest, about the reality, which is we won’t get them all out, but we are doing it every hour of every day to get them through.”
Earlier, armed forces minister James Heappey conceded that the Taliban “gets a vote” on the evacuation deadline, ahead of the group seemingly ruling out a continued presence of British or American troops.
Taliban spokesman Suhail Shaheen told Sky News: “This is something … you can say it’s a red line.
“President Biden announced this agreement that on August 31 they would withdraw all their military forces. So, if they extend it that means they are extending occupation while there is no need for that.
“It will create mistrust between us. If they are intent on continuing the occupation, so it will provoke a reaction.”
Downing Street said the UK will continue its evacuation process “as long as the security situation allows”, when asked about the Taliban spokesman’s remarks.
First Minister Nicola Sturgeon warned again a “cut-and-run operation” in Afghanistan and said Nato countries should remain for as long as required.
Ms Sturgeon said: “I think it is deeply regrettable that the current situation we’re seeing unfold right now has been allowed to develop in the way that it has, but we have to go forward from where we are right now.
“The world has a massive responsibility towards people in Afghanistan and it’s really important that that responsibility is lived up to.”
Asked how many refugees Scotland would accept, Ms Sturgeon said: “I’m hoping those discussions will continue and intensify over the coming days. There’s been some correspondence between myself and Boris Johnson and between my ministers and UK Government ministers.
“We don’t yet have a number that Scotland will welcome here, but I hope to get to that as soon as practically possible.
“Scotland stands ready and willing.
“There’s lots of work to be done – lots of practical arrangements – but as well as living up to our responsibility to give refuge to people fleeing horrific circumstances like those seen in Afghanistan right now, we stand to gain a lot as well.
“Many of the Syrian refugees who came here are contributing massively to Scotland, they’re establishing businesses and working to make a contribution.
“So this is not just one-way traffic, there are lots of mutual benefits here.”
SNP Spokesperson for Defence Stewart McDonald wished the troops luck, but also called for an extension to the August 31 deadline.
He said: “The troops being deployed from 3 Scots carry all of our good wishes, as they enter an extremely complex and dangerous environment to do a tough job.
“I’ve no doubt their professionalism and commitment to duty will serve them well as they do their part in Afghanistan.
“On a political level we need to see the government and other NATO allies do as much as possible to get the 31st August deadline extended. It’s clear from Ben Wallace’s comments that we need more time to help more people and I hope we can see progress on that. We must not turn our backs on the people of Afghanistan.”
Meanwhile, it was reported that a firefight at one of the gates of Kabul’s international airport killed at least one Afghan soldier on Monday.
The Foreign Office said it had sent five extra staff to Kabul airport, taking its total working on the evacuation effort in the capital to 19.
Mr Heappey acknowledged not everybody will be able to be evacuated, as he said there are still “thousands more” people the UK wishes to evacuate, including British nationals.
The defence minister said that the Taliban, which swept to power last week as the US withdrew its troops, “gets a vote” on the evacuation deadline.
“It’s just the reality, we could deny them the vote, we have the military power to just stay there by force, but I don’t know that the humanitarian mission we’re embarked on at the moment which is to evacuate as many people from Kabul as we possibly can is helped by Kabul becoming a warzone,” the minister said.
Mr Biden signalled on Sunday that he did not want US armed forces to stay in the central Asian country beyond August.
Asked about delaying the withdrawal of American troops during a press conference, the US president said: “Our hope is that we don’t have to extend but there are discussions going on about how far we are.”
Foreign Secretary Dominic Raab and Defence Secretary Ben Wallace held talks with their Washington counterparts over the weekend to call for an extension.
Labour leader Sir Keir Starmer has written to the Prime Minister calling for more information on how the UK is planning for the next stages of the rescue mission.
Sir Keir asked whether Mr Johnson had “spoken personally” to Mr Biden to “ask him to extend the evacuation period beyond the end of August”, and whether the UK was working on a contingency plan with Nato allies to “hold Kabul airport without US troops”.
Government officials said there is “no fixed date” on when the UK will withdraw, but it is feared that without US boots on the ground, the remaining allied forces would be unable to secure Hamid Karzai International Airport from the crowds looking to flee the Taliban takeover, or other potential security threats.
The Ministry of Defence confirmed that 5,725 people have been repatriated since rescue efforts began on August 13, with 3,100 of them Afghan individuals and their families.
On Sunday, 1,721 people were airlifted from Kabul by the Royal Air Force across eight flights.