Solomons officials said two 5ft waves hit the western side of Santa Cruz Island, damaging up to 80 homes and properties, said George Herming, a spokesman for the prime minister.
Many villagers had headed to higher ground as a precaution, Mr Herming said.
Solomon Islands police commissioner John Lansley said local patrols reported that several people were presumed dead, although the reports were still being verified.
Four villages on Santa Cruz were hit by the waves, with two suffering severe damage, Mr Lansley said.
Other areas of the Solomons did not appear to have been seriously affected, and authorities cancelled warnings for tsunamis on more distant coasts.
Disaster officials were struggling to reach the remote area after the tsunami flooded the air strip at the nearest airport and left it littered with debris.
The tsunami formed after an 8.0-magnitude earthquake struck near the town of Lata, on Santa Cruz in Temotu province, the easternmost province of the Solomons, about a three-hour flight from the capital, Honiara.