Appointments for many patients had to be rescheduled after Seqirus, the largest provider of flu vaccines to the UK, confirmed delays of up to two weeks in England and Wales.
The company blamed "unforeseen challenges linked with road freight delays" for the disruption.
Professor Anthony Harnden, deputy chairman of the Joint Committee on Vaccination and Immunisation (JCVI), told BBC Breakfast on Saturday: "Clearly influenza immunisation this year is really important and the reason it's so important is because of lockdowns we've had very low circulating influenza levels last winter.
"So we do know when there are low circulating influenza levels the year before, often we get high infection rates in the following year so it's quite possible that we'll have a high instance of influenza this year.
"So it's really important to get the influenza immunisation and really important to get it as soon as possible.
"It is of concern that there are delays, but hopefully this will be corrected soon and we'll get on in general practices, we always do."
An apparent exodus of HGV drivers from EU countries, who returned to the continent during the coronavirus pandemic and remained there, has been blamed for disruption in sectors of the economy in recent weeks.
GP Online reported that it had been sent a letter advising practices not to rebook appointments until they receive confirmation of a new delivery date for vaccine supplies.
Shadow health secretary Jon Ashworth has said: "GPs are at their wits' end, first forced to cancel blood tests because of bottle shortages and now we learn flu vaccines will be delayed.
"This has been a summer of crisis for the NHS with patients paying the price.
"With winter coming and flu resurgence a huge risk, ministers must urgently get a grip."
A Seqirus spokeswoman said: "Seqirus supplies influenza vaccines to all GP practices in England and Wales.
"Due to unforeseen challenges linked with road freight delays, we have informed all our customers of a consequent delay to their scheduled vaccine delivery by a maximum of one to two weeks.
"Seqirus is working hard to resolve the delay to allow customers to reschedule their influenza vaccination clinics."
Dr Richard Vautrey, British Medical Association GP committee chairman, said the issue is likely to affect a "significant proportion" of practices.
He warned of a "serious impact" on practice workloads and patients.
Free flu vaccines will be available to more than 35 million people including all secondary school students this winter, according to the Government.