Officials will visit residents door-to-door in areas under the N3 postcode to deliver PCR test kits and a mobile testing unit will be set up in the car park of Finchley Central Station.
The council said in a statement: “The South African variant of Covid-19 has been found in Barnet.
“From Thursday 15 April we will start testing people for this variant in specific postcode areas affected in N3 or those who shop on the local high street.”
It comes as thousands of people queued to take coronavirus tests across London, after new cases of the variant were found in several boroughs.
Wandsworth Council in south London called the testing operation a “mammoth task” but urged all adults in the area to get tested even if they are not displaying any symptoms.
People in Lambeth and an area of Southwark have also been asked to do the same.
According to the Department of Health and Social Care (DHSC), around 44 confirmed cases of the South African variant of Covid-19 have been found in Lambeth and Wandsworth, with a further 30 probably cases identified.
The BBC reported that the outbreak appears to have been triggered by an individual who travelled from Africa in February, before the country involved was put on the red list for mandatory hotel quarantine.
Documents seen by the broadcaster showed that the individual did self-isolate and requested a home test after showing symptoms, but it is thought that the infection spread to members of their household and then to a care home in Lambeth.
On Wednesday morning, a government adviser said the easing of lockdown restrictions may need to be reversed if coronavirus variants spread rapidly.
Professor Peter Openshaw, a member of the Covid-19 clinical information network, said his fellow scientists were “very concerned” over the cluster of cases of the South African variants detected in London.
He told BBC Newsnight: “I think we’re all just hoping that the staged reduction in lockdown is going to be OK. It is being done reasonably cautiously but I think this is not good news.
“If we get rapid spread of the South African or other more resistant variants, it may well be that we are going to have to put the reductions of lockdown into reverse.”