The London borough is the UK local authority with the highest rate of HIV.
Local councils and charities warn HIV test guidelines may not be implemented in England because of a lack of funds.
New guidance from the National Institute for Heath and Care Excellence aims to increase testing in people with undiagnosed HIV in England.
The guidance is published to coincide with World Aids Day.
It is estimated that 103,700 people are living with HIV in the UK and 17% of people with the virus are unaware of their infection, so risk unintentionally passing it on to their sexual partners.
Part of the new NICE guidance focuses on testing for HIV, which is the responsibility of local authorities, where there are high or extremely high rates of HIV. Two-thirds of late HIV diagnoses occur in these areas.
Over one-third of the 152 council areas have high rates.
The updated guidance recommends all patients in areas with high and extremely high rates of HIV be offered a test on admission to hospital, if they have not previously been diagnosed with HIV and are undergoing a blood test for another reason.
In extremely high rate areas, hospitals should offer the tests even if they are not having blood tests as part of their care.
GP surgeries in high and extremely high-rate areas should also offer patients an HIV test on registration.
NICE also recommends testing community settings in these high rates areas, such as pharmacies, the voluntary sector and venues where there may be high-risk sexual behaviour.
HIV experts have strongly welcomed the new guidance but told BBC's Victoria Derbyshire programme they are concerned the NICE guidance may not be implemented because of a lack of funds.
Dr Chloe Orkin, from the British HIV Association, said prevention was simply not "high" enough on the government's agenda given pubic health budgets are being cut by nearly 4% a year.
Councillor Izzi Seccombe, of the Local Government Association, said achieving what NICE was asking was going to be difficult. "The strain placed on councils by the cuts by central government to public health budgets would make commissioning HIV testing in all surgeries and hospitals in high and extremely high-risk areas an unaffordable burden.
"Despite these limited resources, testing those in high-risk areas must always be a priority. Councils are commissioning HIV testing in a variety of settings."
But the Department of Health maintained councils had been provided with sufficient funding.
Nonetheless, the Elton John Aids Foundation feels councils need help and has offered to fund HIV testing in Lambeth for two years.
David Furnish, chairman of the Elton John Aids Foundation, said: "I believe everyone should have an HIV test. We know we can make a difference in Lambeth, but there is no reason why we can't do this in future in other high-rate areas."
Jennifer Reiter, of Lambeth Council, added: "We value Elton John Aids Foundation's support and are exploring ways with them to increase access to HIV testing."
The BBC's Victoria Derbyshire programme is broadcast on weekdays from 09:00 GMT on BBC Two and the BBC News Channel.
The Elton John Aids Foundation has offered to finance HIV testing in Lambeth, the Victoria Derbyshire programme has learned.