The United Nations on Tuesday expressed serious concern over the deaths of at least 10 people in Kenya, where police and protesters clashed during anti-government demonstrations the previous day.

The violence erupted on Saba Saba Day (meaning Seven Seven) when demonstrators annually mark the events of July 7, 1990, when Kenyans rose up to demand a return to multi-party democracy after years of autocratic rule by then-president Daniel arap Moi.

"We are deeply troubled by the killings yesterday of at least 10 people, as well as looting and destruction of property in Kenya," UN rights office spokeswoman Ravina Shamdasani told reporters in Geneva.

She said that "lethal ammunition, rubber bullets, tear gas and water cannons were used" as police responded to the protests.

She highlighted that Kenyan police had reported that at least 11 people were killed, 52 police officers injured, and 567 arrests made.

The Kenya National Commission on Human Rights, meanwhile, reported at least 10 deaths, 29 injuries, 37 arrests and two abductions, she added.

"We have also received reports of looting and damage to public and private property by unidentified individuals in multiple locations."

Shamdasani said the violence came "barely two weeks after 15 protesters were reportedly killed and many more injured in Nairobi and other parts of Kenya on 25 June".

UN rights chief Volker Turk renews "his call for calm and restraint, and full respect for the freedoms of expression, association and peaceful assembly", she said.

"It is essential that legitimate grievances at the root of these protests are addressed," the spokeswoman said.

The UN rights office noted that Kenyan police had announced an investigation into earlier incidents.

Shamdasani stressed that "under international human rights law, intentional lethal force by law enforcement officers, including with firearms, should only be used when strictly necessary to protect life from an imminent threat".

Turk reiterates "his call for all reported killings and other alleged violations and abuses of international human rights law, including with respect to use of force, to be promptly, thoroughly, independently and transparently investigated", she said.

"Those responsible must be held to account."