Both have been designated variants of concern by health experts.
There is no evidence either of the variants make people more ill, but they are thought to be more transmissible than the variant in the UK at the very start of the coronavirus outbreak.
Current vaccines were designed around earlier versions of coronavirus, but scientists believe they should still work, although perhaps not quite as well, against these variants.
The Department of Health and Social Care (DHSC) said that NHS test and Trace was working with Tower Hamlets Council to provide additional testing and genomic sequencing in targeted areas within the E1 postcode from Sunday.
It follows the identification of "several" confirmed cases of the variant B.1.351, first seen in South Africa, and the P1 variant, first identified in Brazil.
The department said all the confirmed cases were self-isolating and there were no links between the new cases and the cluster of cases recently identified in the south London area.
Everyone aged 11 and over who lives, works or is educated in these postcodes is being encouraged to take a test when invited, whether they are showing symptoms or not, the DHSC said.
The announcement comes as the government said a further seven people had died within 28 days of testing positive for Covid-19 as of Saturday, bringing the UK total to 127,524.
Separate figures published by the UK’s statistics agencies show there have been 152,000 deaths registered in the UK where Covid-19 was mentioned on the death certificate.
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