The 66-year-old made the announcement on Twitter late on Monday but is currently experiencing mild symptoms.
'According to the test results made today, my Corona Virus result was positive,' he wrote.
'I'm in good hands at the hospital. Do not worry. About to communicate as soon as possible.'
Galatasaray also took to Twitter to express their support for Terim, who is in his fourth spell as manager of the club.
'What did we not achieve when we put it shoulder to shoulder! We are with you, teacher! Once again,' a Twitter message from the club read.
Earlier on Monday, Galatasaray announced that the club’s vice-president Abdurrahim Albayrak and his wife had also tested positive for Covid-19.
Terim is one of the most experienced coaches in Europe, having also had spells in charge of AC Milan and Fiorentina.
He has won the Turkish Super Lig eight times in his four spells as Galatasaray manager, and he also led them to the UEFA Cup in 2000.
Terim has also taken charge of the Turkish national team on three separate occasions, leading them to the semi-finals of UEFA EURO 2008.
The Turkish league was only suspended last week following repeated calls to grind it to a halt because of the Coronavirus pandemic, with Galatasaray third on the table, three points off the top.
Turkey has recorded 1,529 Coronavirus cases so far with 37 deaths.