They have released still images from CCTV cameras and the country’s defence minister says that they hope to make an arrest “as soon as possible”.
Yesterday the pair were named by Thai police as 24-year-old David Miller from Jersey and 25-year-old Hannah Witheridge from Great Yarmouth in the UK.
One of the images released by police shows what they said were the two victims walking together hand-in-hand.
They also said a man pictured in a later image, who looked Asian, was considered a prime suspect, but they added they are following a number of leads in the case.
“We are focusing on migrant workers because of the surrounding witnesses and evidence, including the video footage,” Major General Kiattipong Khawsamang said.
“We are sweeping hotels, bars, businesses and residences of migrant workers on the beach to find the suspect.”
Thai police have also confirmed that they are not stopping people coming to or leaving the island, but that they do have officers stationed at the ports.
People on the island of Koh Tao – a divers’ resort popular with backpackers – held a candlelit vigil for the dead pair last night.
The brutal murder has been covered across the international media. Thai Prime Minister, General Prayuth Chan-ocha said: “This should not have happened in Thailand. It will affect our image in the eyes of international countries.”