The Spanish giants are set to compete with Manchester United and Tottenham should the £20million-rated youngster become available.
Bournemouth are also interested in taking the 20-year-old to the Premier League and will offer a direct route to the first team.
But Atletico will signal their serious interest by sending representatives to Saturday's showpiece against Motherwell at Hampden Park.
They will watch the Scotland left-back attempt to complete the treble in a season that has seen interest in him increase.
The Scotland left-back has caught the eye on three seasons since breaking into the first team at Celtic Park.
United are long-term admirers of him but are also looking at established internationals Alex Sandro and Danny Rose for that position.
Tierney admitted just last week he enjoys the attention but is happy to stay at his boyhood club for the long term.
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He signed a lucrative sic-year contract earlier this season to make him one of the Scottish champions' top earners.
And the Mail reported him as saying: "When big teams are looking at you, it's flattering.
"It's happened during most windows since I broke through. It's good for me.
"It doesn't bother me or distract me because I'm enjoying myself.
"I'm playing for Celtic, so I'm happy. I'm not in any rush at all.
"I had four years left on my last contract and, if I wanted to leave, I would have chosen to run that down.
"Instead, when the club offered me six I was happy to sign."
Tierney added: "That's three seasons I've been playing regularly for Celtic and that's a long time.
"But I still see myself as being young and having a lot to learn.
"I've been around the dressing room and seen a lot of people come and go."
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