The testing will be targeted within the postcodes of IP22, SO15 and GU22 in those areas respectively.
Efforts in Manchester to track down examples of the mutation of the more transmissible Kent variant will also be expanded following deployment of testing teams last week.
The postcode districts of M40 and M9 are set to be targeted.
Some 10,000 extra tests were rolled out in areas of the city last week after four people from two unconnected households were found to be infected with the E484K mutation, which is linked to the Kent variant mutation.
Laboratory studies have shown that viruses with the E484K mutation, which is also found in the South African variant, can escape human defences, making them more efficient at evading natural and vaccine-triggered immunity.
People living within the targeted testing areas are strongly encouraged to take a Covid-19 test this week, whether they are showing symptoms or not, a spokeswoman for the department said.
People with symptoms should book a test in the “usual way”, and those without symptoms should visit their local authority website for more information, she added.
DHSC said surge testing in parts of the London boroughs of Haringey and Merton, along with Sefton in Merseyside, were complete.
“Further data on surge testing will be provided in due course,” a spokeswoman said.