Most of the 3,000 French troops part of the 40-nation Nato-led Isaf force are in Kabul province, which includes Sarobi, and also Kapisa province, northeast of the capital.
Around 21 soldiers were also wounded in the battles.
The Taliban movement confirmed that it had attacked Isaf troops in Sarobi and blown up several vehicles.
"We have inflicted heavy casualties," said Zabihullah Mujahed, a Taliban spokesman.
The fighting in Sarobi comes as Nicolas Sarkozy, the French president, is due to arrive in Kabul on Tuesday.
Sarkozy said his visit to Afghanistan is aimed to assure French troops "that France is at their side".
AFP, the French news agency, said Tuesday's deaths bring the number of French soldiers killed in Afghanistan since 2002 to 24.
It is the highest number of fatalities suffered by the French army since 1983 when 58 paratroopers were killed in Beirut.
US base 'stormed'
Dozens of Taliban fighters and suicide bombers also attacked a US military base in eastern Afghanistan, officials said.
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At least 13 people were reportedly killed on Tuesday, some in their own suicide blasts.
Isaf had confirmed that Camp Salerno in the eastern town of Khost, 30km from the border with Pakistan, had been attacked but could give few details.
"We have heard about suicide bombers on foot. They are receiving indirect [rocket and artillery] fire," said an officer in the Isaf media office in Kabul, the capital.
A suicide car bomb at the base on Monday killed 10 Afghan labourers waiting outside.
Security forces were able to prevent a second suicide attack moments later, according to the US-led coalition and Afghan officials. The Taliban had claimed responsibility for that attack.
General Mohammad Zahir Azimi, the Afghan defence ministry spokesman, said 15 "terrorists" had attacked the base on Tuesday.
"Our commando units were deployed into the area and engaged the attackers and pushed them back," he said.
"So far 13 attackers have been killed. Six blew themselves up, six others died in the explosions and one died in gunfire from commandos. Their bodies have been recovered."
Al Jazeera's Zeina Khodr, reporting from Kabul, said that local and international security forces are launching an operation to capture the remaining fighters involved in the attack.
"The operation was launched approximately 12.30 local time overnight by the fighters, and was aimed to storm the military base," she said.
"They were not able to storm the base, and were pushed back, according to the governor of Khost province, and are now said to be hiding out in corn fields and civilian homes."
Conflicting reports
An Afghan army officer said on condition of anonymity that four US soldiers had been wounded. This could not immediately be confirmed by the US military.
Arsala Jamal, the governor of Khost, said his information was that "about 30 Taliban tried to attack the Salerno base".
"They were fired at. We have found six bodies which were all wearing suicide vests," he said.
"Some of them have blown themselves up. Others are hiding in nearby houses and corn fields. The troops are searching for them."
Jamal also said that four Afghan army troops had been wounded in the fighting,
According to an Afghan army officer, the attackers had also launched rockets at the base.
Violence has escalated in the summer and military officials said attacks have increased by 50 per cent up this year compared to last year in some parts of the country.