The Bundeliga giants have been attracted by the left-back’s performances as they monitor their on-loan right-back Jeremy Toljan.
Dortmund, currently level on points with Bayern Munich at the top of the German league, have their finger on their pulse when it comes to young talent in Europe.
Jadon Sancho has become an England international since moving to the club, while Manuel Akanji and Dan-Axel Zagadou are two of Europe’s promising defenders.
They will face competition in the summer to do a deal for the £25million defender with Arsenal and Leicester City both reportedly interested.
Brendan Rordgers is keen on signing his former player if, as expected, Leicester’s England international left-back Ben Chilwell leaves in the transfer window.
Tierney will have seen the success his Scotland team-mate Andy Robertson has had since moving to the Premier League.
However, he has spoken of his desire to ignore all transfer talk and focus all his efforts on Celtic.