The attache, identified as Mikail Gullu, was arrested at an airport in the eastern Saudi city of Dammam as he attempted to board a flight to Germany, the reports said.
"Saudi authorities are holding the military attache for the Turkish embassy in Kuwait," the kingdom's Asharq Al-Awsat daily reported, quoting what it described as a foreign diplomatic source.
"He was stopped based on a Turkish request and was detained during his attempt to flee over his likely links to the coup in Turkey," the daily reported.
Kuwait's Al-Qabas daily said that Gullu had fled Kuwait by land to Saudi Arabia.
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Turkey's ambassador in Kuwait Salih Morat Tamer told Al-Qabas that Gullu is suspected of involvement in the coup attempt but stressed that "accusations against him are yet to be investigated once he is handed over to Turkey."
Judges and military commanders are among 6,000 people arrested over the weekend as Erdogan vowed to stamp out the "virus" of the coup plotters.
Erdogan has blamed Friday's putsch on his arch-enemy, US-based cleric Fethullah Gulen.