Tierney has been relegated to being a backup this season due to the signing of Oleksandr Zinchenko, who is now the Gunners’ first-choice left-back.
Newcastle have had to make do with Dan Burn at left-back this season but are looking to bring in a traditional one in the summer.
They have landed on Tierney as a suitable candidate for that position and could pursue a move for him at the end of the season.
Tierney is not looking unfavourably on the prospect of going to Newcastle and would consider the Magpies should an offer arrive.
Arsenal, on the other hand, are undecided whether to let him leave and are believed to want a sum bigger than the £25m they paid to Celtic for him should they sanction his departure.
Leicester City wanted the left-back in January but the club refused and Tierney himself was happy to stay to help the title effort.
Aside from Newcastle, Tierney has been linked with a host of clubs, including Leeds United, Crystal Palace and Fulham.