Speaking to Championat, the Croatian Under-21 international confirmed there was interest from the Hoops and he was open to swapping Tatarstan for Glasgow.
Uremovic, who can operate at centre-back or right-back, said: "Joining Celtic was a possibility, but it didn't happen. They signed someone else.
"I'm very happy where I am at the moment. Celtic were an option because they play in the Champions League, but the deal didn't take place."
The 22-year-old was linked with a move to the Scottish Premiership earlier this summer, before the Hoops acquired Christopher Jullien from Toulouse for a reported fee of £7 million.
It's believed that Jullien's arrival, or the purchase of Hapoel Be'er Sheva defender Hatem Abd Elhamed, may have ended Celtic's interest in Uremovic.
Hoops boss Neil Lennon is still on the look-out for defensive reinforcements and on top of the ongoing transfer saga involving Arsenal and Kieran Tierney, has seen centre-backs Kristoffer Ajer and Jozo Simunovic linked with moves to Leicester and Lille respectively.
Lennon has, however, ruled out moves for Brentford defender Rico Henry and former England and West Ham forward Andy Carroll.