Agent Megan Carter confirmed that the TV presenter was fighting for her life with her mother at her bedside.
She said: "Charlie has contracted a rare form of malaria and is undergoing treatment at a hospital in Rio de Janeiro.
"Charlie is getting the best treatment available from a team of specialist doctors from Brazil, the USA and the UK."
Charlie's family are flying from the UK to be at her bedside, her spokeswoman has confirmed.
The build-up to the Rio Olympics was overshadowed by concerns about the Zika outbreak, but it is the better known mosquito-borne virus which Webster has contracted.
A friend of hers told the newspaper: "Charlie is battling for survival at the moment - everyone is distraught.
"But she is so strong and a very fit and healthy person in general so we have to keep positive.
"The doctors have now identified the very rare malaria strain she has contracted so she is receiving the right treatment in Rio."
Webster's family, including her mother and brother, have flown to be at her bedside.
She was one of Team GB's official presenters at the Olympics and was working alongside fellow ambassadors for a number of different broadcasters.
The TV presenter is being treated in the intensive care unit at a hospital in Copacabana.
The imposing private hospital is the clinic if choice for Brazil's rich and famous.
Hospital staff declined to discuss Webster's condition when approached for comment.
She arrived in Rio on August 4 and the following night attended the opening ceremony at the Maracana Stadium.
But she is said to have felt unwell during the event and began vomiting. Webster initially put it down to dehydration and felt well enough to post pictures from hospital on social media.
She tweeted: "Very rough day, severe dehydration & infection - 2 drips & antibiotics #Rio."
Later Webster added: "I'm getting there...awful few days with serious infection #Rio".
Last December Webster split amicably from her actor boyfriend of one year Allen Leech, who played Tom Branson in Downton Abbey, with the pair wanting to focus on their respective careers.
Webster was part of a team who cycled 3,000 miles from the UK to Rio over a six-week period to raise money for a cancer charity, the Jane Tomlinson Appeal.
