Already there have been links, some stronger than others, to Arsenal, Barcelona, Leicester City and Lyon.
Last week, reports emerged in France that Tierney is on a three-man shortlist for the French club.
The 22-year-old was named alongside Everton's Lucas Digne and Youssouf Kone of Lille as players who OL would target if Ferland Mendy was sold.
On Wednesday evening Real Madrid announced the signing of the Lyon left-back in a £47million deal.
In addition, veteran Jeremy Morel is out of contract at the end of the month which would leave the club with 30-year-old Brazilian Marcal as the sole player in the position.
Speaking to The Official Ligue 1 Podcast - Le Beau Jeau, Matt Spiro, France-based writer and broadcaster, gave Celtic fans reason to relax somewhat by tipping the club to target the aforementioned Kone who would be the "natural replacement" for Mendy.
However, he did note that the Lille may be reluctant to sell now "when perhaps his value would be even more in a year's time when he will have had a full season in the Lille side".
Spiro said: "The replacements will already have been identified. They are going to have to re-build somewhat the left side of their defence.
"Lyon will have been preparing this for quite some time. They will have money to spend. They have a Champions League campaign to look forward to so it has got to be a priority now replacing Ferland Mendy with a top quality player."