The shortage of nurses worsened last year, with 17% of all London's registered nursing jobs vacant, up from 14% in 2014 and 11% in 2013.
The figure is much higher than the national average of 10%.
The Department of Health said it did not recognise the figures and London had 1,800 more nurses than a year ago.
The RCN said the new figures, which it gathered through Freedom of Information requests to all of London's NHS trusts, showed the city was facing a "critical shortage" of registered nursing staff.
A spokesperson said this put patients at risk and led to expensive solutions such as temporary agency staff or recruitment from overseas.
Bernell Bussue, the RCN's London regional director, said: "The problem is partly down to short-sighted workforce planning which saw training posts cut in the past, meaning there aren't enough home grown nurses coming through the system.
"Most importantly, the ongoing pay freeze imposed by the government means that nursing staff increasingly just can't afford to live and work in London.
He urged the government to give nurses a pay rise so they could settle in the capital, saying nurses' pay had "run 10% below inflation since 2010".
10,140
total vacancies
17% average vacancy rate
30% highest vacancy rate at an individual trust
3% lowest vacancy rate at an individual trust
Responding to the figures, London Mayor Boris Johnson told BBC Radio London "you can afford to be a nurse and live in London".
He added: "I won't deny the cost of living in London is incredibly high."
But he said the Conservative party has built record numbers of "affordable homes" and there are homes for part-buy, part-rent.
A spokeswoman for NHS England (London) said the July figures did not reflect reductions in agency spend since a cap on charges for agency staff was introduced in November.
"In London we are looking at new ways to recruit both new and returning nurses while retaining nurses already in post so that we are reaching our planned staffing levels," she said.
"This includes a programme in which senior nurses in the capital are working together to  create innovative career pathways and making London a more desirable place to work."
A Department of Health spokesperson said: "Official statistics show that Londoners have already benefitted from 3,400 additional nurses since May 2010 and this is down to continued government investment in the frontline.
"We have 50,000 nurses in training and our recent changes to student funding will mean up to 10,000 more training places across the country by 2020."
More than 10,000 vacancies for nursing posts in London went unfilled in 2015, new figures from the Royal College of Nursing have shown.