According to a report in the Daily Telegraph, rioters at one of the company's Kenyan farms burned vehicles and its storage facility and tried to burn the tea plants.
There was also heavy fighting in the Kericho area.
"We think 11 people have been killed, eight of whom are our current employees," Trevor Gorin, a spokesman with the firm, told FoodProductionDaily.com.
The company had already ceased normal operations and is now moving people to safer locations.
According to the newspaper report, tea companies have begun evacuating some 30,000 displaced people.
"We've not had any significant business impact so far but the situation remains very difficult," added Gorin.
Kenya is the world's third-largest exporter of tea.
Tea auctions in Mombasa have been suspended since rioting broke out following presidential elections in the country on 27 December.
Major roads leading to Mombasa, the country's second city and the region's key port, have been blockaded and there is also a fuel shortage in many places.
Transporting goods is also made difficult as firms are unwilling to risk damage to vehicles and workers caused by rioters.
Unilever's Kenya estates produce around 30,000 tons of black tea each year, accounting for 10 per cent of national production.
The firm also sources from other locations, mainly using Kenyan tea in blends.
The impact of ceasing operations in Kenya is not yet clear, added Gorin, and will depend on how long the current situation lasts.
Finlay's Tea, which also runs a tea estate in the Kericho area, is also facing disruptions with no tea leaving its farm for several days.
Unlike Kenya's other major export, fresh fruit and vegetables, tea is not perishable so supplies can be held at source for some time.
Kenya-grown fruit and vegetables seem to be reaching the UK however, with no reported problems with air freight.
Major supermarkets including Asda, Sainsbury's and Marks and Spencer all said they experienced no disruptions to fresh produce supply through last week.
Kenya produces significant quantities of green beans as well as other vegetables such as aubergines, chillies and prepared salads.
"We will continue to monitor the situation and work closely with our suppliers in Kenya ," said a spokeswoman at Asda.
Flamingo Holdings, a major supplier of Kenya-grown fruit and vegetables to British supermarkets, said violence in the country has not impacted the key horticultural areas.