Manchester United are being linked with transfers in and out in their left-back position.
The Mirror add that Tierney is valued at £10m and is also being monitored by Chelsea, Arsenal and Manchester City.
Shaw and Tierney are very similar.
The British pair are both very attacking full-backs, with pace one of their key attributes.
England international Shaw, 21, and Scotland star Tierney, 19, also have injury problems in common.
Shaw missed the best par of a year with a broken tibia recently and has also batted groin, hamstring and ankle issues during his fledgling career.
Meanwhile, Tierney has missed just over two months of this season so far with damaged knee ligaments.
Tierney made his Celtic debut back in 2015 and has been a regular first-team player for the past one and a half seasons. Despite his important role, he still wears no.63 on the back of his shirt.