The first quake on Tuesday afternoon hit offshore at a depth of 15 kilometres about 16 km southwest of the area of Malango, said the United States Geological Survey, which initially put its magnitude at 7.3 before revising it down to magnitude 7.0.
A second quake, with a magnitude of 6.0, struck nearby 30 minutes later.
"There are no known injuries but the roof of the High Commission annex has collapsed, which would point to likely damage throughout the city," Australian Prime Minister Anthony Albanese told parliament.
Honiara International Airport suffered damage to its ceiling but the building was intact, a Solomon Islands Airline worker at the airport told Reuters by phone.
Aftershocks continued to be felt, he said, declining to be named as he was not authorised to speak publicly. Airport staff would continue working but the damaged section of the airport terminal was closed to passengers.
The Solomon Times newspaper reported power had been cut for most of Honiara, as preliminary assessments of damage to power lines are made.
The Solomon Islands Meteorological Service said there was no tsunami threat but warned about unusual sea currents.
"People are also advised to be vigilant as aftershocks are expected to continue," an employee said on social media.
Widespread power outages are being reported across the island and the Solomon Islands Broadcasting Corporation said on Facebook that all radio services were off air.
The National Disaster Management Office said it has received reports that people felt the quake but were waiting for reports of damage.
"People in Honiara moved up to higher ground in the minutes after the earthquake but some have now moved down," an official told Reuters by phone.
