The National Disaster Operation Centre said the first blast occurred on a 14-seater matatu, or public minibus, and the second was inside Gikomba Market, situated to the east of Nairobi's central business district.
A spokesman at the Kenyatta National Hospital, Nairobi main hospital, said eight bodies had brought in and "more than 70" people admitted for treatment, many of them in a serious condition.
"Many of the injured are bleeding profusely. We need a lot of blood," the spokesman, Simon Ithae, told AFP.
Kenyan media reports said bombs were thrown at the minibus and into a shop in the market. Images showed a red minibus with its windows blown out and the surrounding area strewn with debris.
The attacks come amid a wave of bombings and attempted bombings in both Nairobi and the port city of Mombasa, and come days after several countries issued new warnings against travelling in Kenya.
Yesterday and today, hundreds of British tourists were being evacuated from beach resorts near Mombasa because of the travel warnings.
Earlier this month three people were killed and 86 wounded in twin bus blasts in Nairobi that were blamed on Islamic militant cells linked to Somalia's al-Qaida-linked Shebab rebels. The previous day twin attacks left four dead in Mombasa.
Kenya has been targeted by the Shebab since sending troops to war-torn Somalia in 2011. Kenyan soldiers are still posted in southern Somalia as part of an African Union force supporting the country's fragile internationally-backed government.
There was no immediate claim of responsibility for the latest attacks — although Kenyan authorities have been engaged in a major security crackdown on suspected Shebab supporters in Nairobi.
The operation has focused on Nairobi's main Somali district Eastleigh, and residents have accused police of indiscriminately arresting people of Somali origin. Gikomba Market is situated adjacent to Eastleigh.
The Shebab claimed responsibility for the high-profile attack on Nairobi's Westgate shopping mall last year in which at least 67 people were killed.