The Circle and Hammersmith & City line will be closed this weekend due to more than 300 staff self-isolating, according to Transport for London.
There will also be changes to services on the District and Metropolitan lines.
TfL blames a shortage of control room staff and warns of further disruption on other parts of the Tube network because of ‘late-notice staff shortages’
More than 600,000 people have been told by the NHS Covid app to self-isolate in recent weeks, leaving industries across the UK struggling to stay open - and pushing the government into releasing a list of critical workers who could be exempt from quarantine.
Environment Secretary George Eustice earlier accepted trains are likely to be cancelled and delayed in the coming weeks due to the "pingdemic".
The exemptions list, which applies to some workers within 16 industries, has been criticised as being confusing, too narrow, and for failing to cover a wide range of workers many would consider to be critical.
Mr Eustice defended the list saying it is intentionally "quite limited" because the government wants to avoid a "general approach that anybody can say they're critical".
"It's been deliberately drawn to be quite narrow," he said.
Asked how train services could continue with only a few roles within the industry protected, Mr Eustice said: "You have to differentiate between potentially having a lack of train drivers in which case not every train runs on time and some trains are cancelled and that's a consequence of staff absence.
"We recognise that that's a reality over these next few weeks," he added.
Rail unions say train services will be reduced from next week because of staff shortages, which they warn could lead to overcrowding.
Thameslink and Southern will cut its weekday timetables on five routes from Monday "until further notice".
Avanti West Coast is also cutting trains from Monday on its routes between London Euston and Manchester, Birmingham and North Wales.
London Northwestern Railway will introduce a revised timetable with fewer services from Saturday, while South Western Railways said it had "made pre-emptive changes to some services this weekend".
"Other services may also be subject to short-notice change," it added.
Mick Lynch, general secretary of the Rail, Maritime and Transport union, said: "There is a real danger of a headlong rush into these new measures driven by this inept government which could make a bad situation even worse.
"We are already hearing of planned reductions to rail services next week due to staff shortages."
"Before any implementation of new procedures, employers need to produce proper risk assessments agreed with the union that consider and control the enhanced risks to all staff and ensure that the principle objective of workplace safety is maintained and fully enforced."
Numerous unions are calling on the government to widen the exemption list to more workers within their industries.
At present, the policy will only apply to named workers if their employer has received a letter from the relevant government department - those employees must also have received two coronavirus vaccinations.
If a named employee is told to self-isolate by either NHS contact tracers or by the NHS Covid app, they will be able to avoid quarantine with a negative test result - anyone who tests positive will still be expected to self-isolate.
The list came as it was announced that almost 620,000 people in the week to July 14 had been told to self-isolate by the NHS Covid app.