Boris Johnson has threatened to “privatise the arse” off the Passport Office if it cannot clear its backlog and deliver better value for money, a senior government source has told reporters.

The prime minister also put the Driver and Vehicle Licensing Agency (DVLA) and other “arm’s length” bodies on notice in comments to cabinet ministers, according to the source who attended Tuesday’s meeting.

There had been an unprecedented surge in demand after the lifting of coronavirus restrictions because 5 million people had delayed renewing their passports during the pandemic.

Under post-Brexit EU travel rules, UK citizens must have at least three months’ validity remaining, which threatens to put more strain on the system.

It comes amid fears of summer of travel chaos with Brexit queues at Dover and concern about the routine cancellation of flights.

Johnson is understood to be “horrified” at the backlog in dealing with passport applications and will summon the agency’s leadership for urgent talks at No 10 next week.

He is said to be concerned that families wanting to go on summer holidays are under pressure to pay up to £100 extra per passport for fast-track applications because of delays.

Pressed on whether privatisation was an option for the Passport Office, Johnson refused to rule it out in an interview with TalkTV. “I’m not going to rule anything out. I don’t mind if it’s public or private, what I want is to deliver value for money and help keep people’s costs down,” he said.

“When I see institutions not delivering things like passports or driving licences in a speedy way ... we want action.”

The source cautioned that privatisation would not be the starting point, but warned that Downing Street could look to businesses to take over if passport bosses could not meet the 10-week target for standard applications costing £75.50.

Johnson was also said to have criticised “post-Covid mañana culture” at some of the bodies for not adapting after the easing of coronavirus restrictions.

As well as the Passport Office, MPs have been raising concerns over significant delays in turning round applications for licences at the DVLA.

Heathrow Airport’s chief executive, John Holland-Kaye, said it was “vital” that people could go on holiday this summer in case the UK reintroduced coronavirus travel restrictions.

“Normally the Passport Office is very slick in processing passports. It’s really disappointing to hear [about delays].

“If it’s not resolved, it is going to mean that people can’t get away.

“After two years of lockdown, we need to make sure people can have a good holiday because many of them have vouchers that they’ve been saving up, they’ve got trips of a lifetime that they’ve put on hold, and we don’t know when things might get closed down again.

“Making the most of the summer is vital,” he said.

On Monday, the Commons heard about one woman who had waited more than five months to receive her daughter’s new passport, and services were branded “either really very good or an absolute shambles”.

The SNP home affairs spokesperson, Stuart McDonald, said: “All our constituents are having to cancel holidays, miss funerals, rearrange visits, with even a new 10-week target routinely being failed.

“What will be done to avoid this predictable mess getting worse? And can we be assured that the 10-week target will not be lengthened further as we approach the summer?”

Home office minister Kevin Foster said: “We don’t have any intention to further extend that standard.

“We are at the moment processing most passports well within it, but we would advise people this is a very, virtually unprecedented surge in demand, and if people are planning to travel this summer we would advise them to get their application in as soon as possible.”

He said the Passport Office dealt with a million applications last month, compared with 7m a year before the pandemic.

Conservative MP Simon Hoare described the backlog as “unprecedented, true, but foreseeable, absolutely”.

The PCS union, which represents Passport Office and DVLA staff, said there is a backlog of applications owing to lack of staff, poor management and failures with private contractors. HMPO planned to recruit 1,700 staff, with 450 in situ by the end of 2021. Only 300 were in place by the end of the year, and they were agency staff, the PCS union said.

The union’s general secretary, Mark Serwotka, said: “It is clear to our members that a crucial service like the Passport Office needs to be properly resourced. It is absurd for Boris Johnson to threaten to privatise HMPO, when it is clear that the current problems are mainly down to increased casualisation of the workforce.

“Many people are desperate to now start travelling again, and so I would call on the Home Office to fulfil their duties to the UK public and adequately fund the service, bringing the whole process back in house.”

TUC General Secretary Frances O’Grady said: “Privatising these services would force people to pay a premium for private profit, [adding to the cost of their passports and driving licenses]. That’s the last thing working people need in a cost of living crisis. Boris Johnson and his ministers are responsible for running these services. Instead of making bad policy on the hoof, he should call in his ministers to get to the bottom of why they are doing such a bad job.”