A small number of petrol stations have been affected, and ministers say people should carry on buying petrol normally.
Earlier, Transport Secretary Grant Shapps told the BBC nothing was being ruled out to help deliver petrol from refineries to stations.
Some ministers believe a temporary work visa scheme to bring more HGV lorry drivers into the UK from the European Union would help.
But others believe such an approach is not a long-term solution to the driver shortages.
A “handful” of BP stations, and a small number of Esso-owned Tesco Alliance stations, were closed on Thursday because of the lack of delivery drivers.
But Mr Shapps said refineries had “plenty of petrol”and that motorists should “carry on as normal”. He told the BBC that the government could bring in the army to drive fuel tankers if it would help.
Despite reassurances, customers have been pictured queuing at some petrol stations.
It is estimated that the UK is short of about 100,000 HGV drivers – with gaps made worse by the pandemic and Brexit.
Reports in several newspapers suggested that the government was considering getting soldiers to drive fuel tankers under emergency plans.
When questioned about this on BBC Breakfast, Mr Shapps said: “If it can actually help, we will bring them in.”
However, he said there would be “technicalities” as to whether military personnel could switch to driving civilian vehicles.
The AA said that most of the UK’s forecourts were working as they should.
“There is no shortage of fuel and thousands of forecourts are operating normally with just a few suffering temporary supply chain problems,” said AA president Edmund King.
“Fridays and the weekend always tend to be busier on forecourts, as drivers either combine filling up with shopping runs, prepare for weekend trips or refuel for the start of the new working week.”
Mr King said drivers should not fill up outside their normal routines, because even if the occasional petrol station was temporarily closed, others just down the road would be open.
“It is now clear that there have been occasional delays over recent weeks that have been managed with hardly anyone noticing. This was a manageable problem.”
There are about 8,380 petrol stations in the UK, according to the Petrol Retailers Association. Of these, about 1% are believed to be closed at the moment.
It is understood that about 50 to 100 BP forecourts are affected by shortages, while Esso said “a small number” of its Tesco Alliance retail sites had been hit.
The Road Haulage Association (RHA) has urged the government to relax visa restrictions for foreign workers in the short term to ease the driver shortage.
But Mr Shapps told the BBC he did not want UK lorry drivers to “drop out” because of being “undercut” by cheaper EU labour.
He said there were “systemic problems” in the haulage industry that needed to be resolved, pointing out that the profession was “99% white male” with an average age of 55, facing poor conditions and wages.
Now, he said, pay was rising and there was “a gradual increase” in new drivers.
While the UK does not lack petrol and diesel at refineries, getting it to forecourts has been difficult in some areas.
It is the latest problem caused by driver shortages, which have already led to supply squeezes in other industries, including supermarkets.