The 39-year-old is currently without a club after leaving Leicester City at the end of last season and was heavily linked with Celtic following Joe Hart 's retirement in 2024.
Celtic ultimately opted to bring in Kasper Schmeichel as their new No.1, while the subsequent arrival of Sam Johnstone from Wolves has left Martin O'Neill well stocked between the posts.
Begovic has now confirmed there was genuine interest from Celtic in the past, while revealing he also held discussions with Rangers.
Begovic told Hajper: "Celtic definitely interested me. I've had some conversations in the past with both Celtic and Rangers, but unfortunately a move never materialised.
"There are many factors in making deals happen, but you can't ignore how big those two football clubs are in terms of tradition. I actually love the SPFL in general, it's a very competitive league and the intensity is always there.
"It's something that we in England really relate to and respect. It never worked out, and while I think time is running out for that to happen, I have huge respect for both clubs, their tradition, and their history."
Begovic also worked under former Celtic managerial target Eddie Howe during his time at Bournemouth , with the Englishman again being linked with the Parkhead job once O'Neill eventually steps aside.
Howe came close to taking charge of Celtic in 2021 before talks collapsed and Ange Postecoglou was subsequently appointed. He is now available again after leaving Newcastle United in July, having guided the Magpies to EFL Cup glory in 2025.
However, Begovic believes his former manager could have his sights set on one of football's biggest jobs.
He added: "Eddie Howe to Celtic? I think Eddie Howe will be looking at some of the biggest jobs in the world, and that's probably where he sees himself.
"What he did at Newcastle is absolutely incredible, he left at a good time and his stock is at a really high point. He's always been linked with the England job, rightly so, and I could see that happening down the line. Anytime a massive job comes up, he has to be in the running.
"He's in for a massive job now, whenever he's ready, rested, and good to go once again."
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