Heathrow Airport regularly sees queues of three hours and sometimes six hours at border control, according to Emma Gilthorpe, its chief operating officer.
Earlier this month, passengers complained of waiting up to seven hours, which one said was "inhumane".
But Border Force said many passengers were arriving without having bought mandatory Covid testing packages.
The enforcement body added that people should only be travelling for limited reasons.
Separately, hundreds of Heathrow passport control are set to take industrial action.
Unions representing Border Force officials said the delays were partly caused by Covid restrictions requiring immigration officials to work in a bubble of 10.
They said this prevented more staff being deployed if the border was particularly busy.
But Nick Jariwalla, director of Border Force at Heathrow, said: "We are in a global health pandemic - it is illegal to go on holiday and people should only be travelling for very limited reasons.
"Unfortunately, a large number of passengers are continuing to arrive without having purchased their mandatory testing packages for the second and eighth days of quarantine in the UK.
"This causes delays as passengers may receive a fine or be subject to other enforcement action."
Mr Jariwalla said all arriving passengers had to purchase the testing package before leaving the airport.
"We make absolutely no apologies for this," he added. "Every essential check stops the spread of coronavirus in the UK.
"These people should not have been allowed to travel without their testing packages and we are following up with regulators and carriers to ensure the law is enforced."
Speaking to the Home Affairs Select Committee, Ms Gilthorpe said that pre-Covid, the queues for EU arrivals were supposed to be 25 minutes, and 45 minutes for non-EU arrivals.
While she said she expected the length of waiting to go up during the pandemic, Ms Gilthorpe said the levels the airport was currently seeing were "unacceptable".
While Ms Gilthorpe said the measures in place for quarantine hotels were running relatively smoothly, with a small number of passengers, she said the same could not be said for the border.
"We are seeing significant pressure on the border and we are seeing very long queues, and that is a worry," she said.
Although Ms Gilthorpe said she knew additional resources had been introduced, the issues were caused by "the level of complexity of the processes and the way resource is being deployed".
International arrivals must now provide proof of a negative Covid test taken 72 hours before departure to England and a passenger locator form which contains details of either a purchased hotel quarantine stay, or two test kits for at home quarantine.
She told the committee: "It is deeply frustrating as the operator of the airport when you have a queue full of people and you only have two desks open.
"It is rare to see all the desks manned and we have to find our way to how we make that happen so we can get that flow."
Heathrow's self-service e-gates at passport control are currently closed.
Ms Gilthorpe said she hoped that by the summer, the gates would also be able to scan passenger locator forms and pre-departure testing results, but that had not yet been integrated into the system.
The Heathrow executive also warned that if the long queues continued after international travel is permitted. it could risk the country's economic resurgence.
"Heathrow is Britain's hub airport, it is the front door. We have to get our economy moving and we have to make sure we are capable of receiving people," she said.
"If you have a poor experience at the border, there's a risk you won't come back again. That traffic will go to Charles De Gaulle [airport in Paris], it will go to Frankfurt and we will miss out on that economic resurgence," she said.
The queue delays come as staff in the PCS union served notice of industrial action over rosters they describe as "unworkable".
This will involve a month of action short of a strike by hundreds of staff, including working to rule and an overtime ban.
The roster changes include ending the ability of staff to swap and request certain shifts, the union said.
In February, staff voted for strike action over the changes, the union said.