Blood and body parts could be seen near the charred remains of a bicycle and a police car at the site one of the explosions.
Neave Barker, Al Jazeera's correspondent in Moscow, said: "This is certainly an indication of another significantly sized attack by insurgents against the pro-Kremlin authorities in the Caucasus."
Grozny, the Chechen capital, has seen a series of bombings in recent months, ending years of relative calm in the city during a rebuilding campaign by Ramzan Kadyrov, the regional president.
The attack comes days after a suicide bombing in neighbouring Ingushetia killed at least 20 people.
Our correspondent said the authorities appeared to be drawing a link between Friday's attacks and the Ingushetia attack.
"The feeling in Moscow from the Kremlin and Kremlin-backed authorities both in Chechnya and Ingushetia are that terrorist attacks in the troubled region are certainly on the rise," he said.
"This is causing a tremendous amount of concern.
"The indication is that in the vacuum after Russian forces left that it appears there is an upsurge in violence across the entire region."
Russia earlier this year ended a 10-year "counter-terrorism" operation in Chechnya, which has been the site of two separatist wars since the 1991 collapse of the Soviet Union.
However, Chechnya and other parts of the Russian Caucasus remain the site of a deadly insurgency led by fighters battling pro-Kremlin local authorities, who in turn have been accused of human rights abuses.