Meanwhile, preparations are underway in Tonga with the capital, Nuku'alofa, sitting directly in Tino's path.
The cyclone, which was upgraded to a category one early on Friday morning, was strengthening as it sat the south of Fiji's western Rotuma island, the Fiji Meteorological Service said.
But it was gathering strength as it moved southeast, where it was heading straight for Fiji's second-largest island, Vanua Levu.
Warnings had been issued for much of the country, with torrential rain, strong winds, and damaging swells forecast.
Its eye was forecast to pass close to Vanua Levu's main town, Labasa, on Friday afternoon, said Stephen Meke, a forecaster with the Meteorological Service.
"We have told them to be prepared, the Ministry of Education has shut down all schools, the evacuation centres are open as well. So everyone, they are aware that there's a system around," he said.
After crossing Vanua Levu, Tino would continue to strengthen to a category two late on Friday as it crossed the Lau group towards Tonga's southern islands.
In Tonga, preparations were underway for its arrival on Saturday, authorities said, with the main island, Tongatapu, in the firing line.
It's nearly two years since Tongatapu was devastated by the category five Cyclone Gita, with the recovery from that still underway.
Mr Meke said while Tino would not have the same strength of Gita (it's forecast to remain a category two), it could still be quite damaging.
"With the cyclone coming down there's also an associated convergence zone that's coming with it, and that convergence zone should bring a lot of damaging gale force wind over the Tongas and it could even extend to Niue," he said.