A picture of a passport circulated on social media, identifying the arrested man as being from the troubled Chinese region of Xinjiang, home to Muslim Uighurs, but Prawut said that picture had not been released by the government and he did not know where it had come from.
The arrest came after security forces detained another foreign suspect in weekend raids on the outskirts of Bangkok, in which they also found explosives.
The man arrested on Tuesday was detained in Sa Kaeo province to the east of the capital near the Cambodian border. He has been transferred to Bangkok for questioning.
Pictures of the suspect shown on television showed a thin man in a baseball cap, sunglasses and with a short moustache.
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Police issued three more arrest warrants on Tuesday for foreign male suspects, Prawut said, adding that one of them may be Turkish.
Police also said they had transferred 22 officers from their posts for negligence. The transfers came just a day after the police chief promised a reward to investigators for making the first arrest.
Police have been criticised for an erratic investigation, which had made little progress until the weekend. Police issued arrest warrants for a Thai woman and a foreign man on Monday.
The woman, Wanna Suansuant, has contacted authorities and would meet police, Somyot said on Tuesday. He gave no details. Her family told police on Monday she traveled to Turkey to work with her partner and child two to three months ago.
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Thai authorities have been looking into a possible Turkish connection. Fake Turkish passports were seized and police requested a Turkish translator to assist in questioning the 28-year-old man they arrested on Saturday.
The authorities have not confirmed his identity or nationality. He is charged with possessing illegal explosives.
Speculation has centered on sympathisers of Uighur Muslims, opponents of the government, southern ethnic Malay rebels and foreign extremists, among other groups.
Thailand drew international outrage in July when it forcibly repatriated more than 100 Uighurs to China.