The 59-year-old, originally from Leigh-on-Sea, Essex, has been unaccounted for since the UN headquarters collapsed on Tuesday.
Ms Barnes's sister Irene Marquet said the family feared the worst.
"There's been absolutely no trace, which is horrendous," she said.
"One wants to remain hopeful but it gets more and more difficult as time goes on."
The former British Airways stewardess has worked for the UN for more than 20 years and has been in Haiti for about two years.
The UK ambassador to the Dominican Republic and other consular staff have gone to neighbouring Haiti.
A British search and rescue team - made up of 71 people, mostly from the fire services, and including two search dogs - flew to the region on Thursday.
Forty of them are working alongside Chinese rescuers in the Carrefour district on the edge of the capital, Port-au-Prince.
The Department for International Development said they had UN protection.
Another group was searching a ruined church in the city centre where a child was thought to be trapped.
The DEC, an umbrella organisation that co-ordinates responses to major disasters overseas, thanked the British public for their contribution to its quake appeal so far.
AT THE SCENE
David Loyn, BBC correspondent, with a rescue team in Haiti
Food, water, medical supplies and shelter are needed on a huge scale for several months to come.
Rescue teams from a dozen countries are doing what they can in the capital, which was the epicentre of the earthquake. A Spanish team that arrived on Thursday morning had pulled out five people by nightfall including a two-year-old boy.
Although the team did not have any of their heavy equipment, which is still stuck in the gridlock at the airport, they dug with simple tools.
Annika Coll, the head of the team, said that "goosepimples went up on my arm" when the boy answered when she called out. He was unharmed after 50 hours without food or water.
The rescue teams know that there could be many more people like that under the rubble and are frustrated by being unable to get out to work.
There are also security concerns. Some 9,000 UN military and police personnel have been trying to keep order, and all rescue teams have had armed guards.
But it appealed for more financial aid in a BBC One TV appeal presented by Kirsty Young.
She said the small Caribbean nation had suffered devastation "beyond measure" and added: "Aid agencies are worried if they don't deal with problems like sanitation immediately, disease will break out and many more people will die."
Mariella Frostrup also presented an appeal on BBC Radio 4.
The Queen donated an undisclosed amount to the DEC and the Prince of Wales gave a private sum to a Red Cross fund.
DEC chief executive Brendan Gormley said: "Efforts on the ground have been hampered by a lack of power and communications problems after the devastating quake but aid is starting to get through and DEC members are working hard in the field.
"It is vitally important that people continue to donate."
The £2m figure for online giving was hit at around 0800 GMT on Friday, and donations by other means are not included.
Money donated to the DEC - which brings together 13 major British-based charities - is being spent on search and rescue, medical care, food, clean water, temporary shelter and clothes.
The prime minister, who visited DEC staff at their north London headquarters on Friday, reassured the public their money would get through.
Shelane Chapman, from Downham in Kent, is one of the 1,000-strong Haitian community in the UK. She has no idea what has happened to her family.
She heard rumours that her mother's house collapsed in the earthquake, killing her six-year-old cousin, but has been unable to verify the news.
Sanitation equipment on its way to Haiti
Mrs Chapman said: "We're sick with worry because we can't get through at all. We keep redialling my mother's mobile number but it just won't work. If the house has gone, where could she be? Where could they all be?"
Judith Craig of United Haitians in the UK, says the charity has seen its membership jump since the earthquake hit.
She attributes the increase to people wanting to find out more and seek solidarity with others.
The Red Cross has set up a
to help people abroad to try to contact their relatives. A similar
has been created by two Haitians in the US.
The Disasters Emergency Committee is co-ordinating an appeal to help the people of Haiti
There are 13 charities involved including the British Red Cross, Islamic Relief and World Vision
Donate via the
or by telephoning 0370 60 60 900
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