The soldiers were detained during a huge search operation in the area around the Aegean Sea resort of Marmaris targeting those behind the attack. Four more soldiers are still on the run, the report said.
The three were detained during a traffic check in the Cetibeli district of Marmaris and have been taken to a police station for questioning.
The attack on Erdogan's hotel, where he was vacationing with family members including his son-in-law, Energy Minister Berat Albayrak, came shortly after he had left the resort by plane for Istanbul.
The president has said that if he had waited 15 minutes more, he would have been killed or taken hostage.
The Dogan news agency published a picture of the three in the back of a police van, looking bedraggled after 10 days on the run.
Turkish Airlines fires 211 staff
Turkish Airlines said Monday it had fired 211 employees over suspected links to US-based preacher Fethullah Gulen and behaviour "conflicting with the interest of our country" in the wake of last week's coup.
The flag carrier said their contracts were terminated due to "the non-fulfillment of performance criteria and in line with the necessary actions we are taking against the FETO structure, attitudes and behaviour conflicting with the interest of our country and company".
The authorities accuse 75-year-old Gulen of running a group it labels the Fethullah Terror Organisation (FETO) that was behind the coup, and Ankara has demanded that Washington extradite him.
The preacher -- who lives in a secluded compound in rural Pennsylvania and whose foundation runs a global network of schools, charities and media interests -- has strongly denied the accusations against him.
In a statement signed by the airline's media relations unit, Turkish Airlines stressed that it stands "united with all of the heroic and honourable Turkish people" against the coup plotters' "malevolent illegal attempt".
"Under any circumstances, we have and will continue to fulfil our responsibility to contribute to democracy."
42 journalists detained
Turkey ordered the detention of 42 journalists on Monday, broadcaster NTV reported, under a crackdown following a failed coup that has targeted more than 60,000 people, drawing fire from the European Union.
The arrests or suspensions of soldiers, police, judges and civil servants in response to the July 15-16 putsch have raised concerns among rights groups and Western countries, who fear President Tayyip Erdogan is capitalising on it to tighten his grip on power.
EU Commission President Jean-Claude Juncker questioned Ankara's long-standing aspiration to join the EU.
"I believe that Turkey, in its current state, is not in a position to become a member any time soon and not even over a longer period," Juncker said on French television France 2.
