The 7.8 magnitude quake sparked a tsunami warning across a wide area of the South Pacific after it struck at a depth of about 49km at 4.38am on Friday.
But in updated advice, the warning centre said all available data indicated the tsunami threat had largely passed.
The US Geological Survey said some casualties and damage were possible from the quake, while the Pacific Tsunami Warning Centre said some coasts in the Solomons, Vanuatu, Papua New Guinea, Nauru, New Caledonia, Tuvalu and Kosrae could be affected within the next three hours.
Oxfam's Abimbola Akinyemi in Honiara told SBS there were unconfirmed reports that a wave damaged 10 houses in a south-eastern province.
"About two hundred metres inland, about 10 houses are damaged, we don't know if they're sunk or washed away but we know that the damage is widespread," he said.
"We don't have information at the moment on the impacts in terms of disruption of lives and property."
The epicentre of the quake, which hit at 4:38 am (1738 GMT Thursday), was located 68 kilometres west of Kirakira, a provincial capital in the Solomon Islands, at a depth of 48 kilometres, USGS said.
“Communities we have made contact with say a small tsunami was generated and people have evacuated to higher ground,” said Solomon Islands director of the National Disaster Management Office Loti Yates.
“We made contact with a health centre on the south coast of Marika.
“There are reports of a collapsed building in South Malaita and there could be more damage to infrastructure.”
“Info coming in is very sketchy. We are not getting any good picture of what is happening,” Mr Yates told SBS before the phone line cut out.
The Pacific Tsunami Warning Centre said waves reaching 1-3 metres above tide level could occur along parts of the Solomons.
The capital Honiara has been blacked out by the jolt, including Henderson International Airport.
The national broadcaster SIBC is also off the air.
The quake could have a significant impact, John Bellini of the Solomon Islands meterological Service said.
"Because it was a large size, it has the potential for some damage. We don't have any specific reports of that at this time, but it is located relatively close to one of the islands there, I'm not sure the infrastructure situation, but it does have the potential to cause some damage."
Freelance journalist Dorothy Wickham in the Solomons capital Honiara says the earthquake woke her up and the aftershocks lasted for several minute
"Oh yes, it was very strong, a few people we called around had things falling, you know, inside their house and breaking.
“I thought, you know, maybe it was just something, somebody walking around that was making the house shake but then I woke up and it was pretty strong, you couldn't stand up properly, it was quite strong.
“And like I said, when I stood up I had to sort of had to stand up properly to keep my balance so it was pretty strong,” she said.
“It was heavy, enough to wake me up. I think it is quite strong,” said Cameron Vudi, Disaster Risk Manager with the Red Cross in Honiara
“I can’t say what’s the damage. Communications difficult with the closest island to the centre.
“We hope to know more in the next few hours.”
In 2013, the islands were hit by a tsunami after an 8.0-magnitude quake, leaving at least 10 people dead and thousands homeless after buildings were destroyed.
Australia's Bureau of Meteorology's Joint Australian Tsunami Warning Centre said there is no tsunami risk to the Australian mainland, islands or territories.
New Zealand Civil Defence said there's no threat to New Zealand.
Authorities are looking to see if there is a tsunami threat to Hawaii, the PTWC said, adding that the estimated earliest arrival of any tsunami wave there would be 2:56 pm Hawaii Standard Time (11:56 AEST).
