The Scotland international has struggled for game time since the arrival of Oleksandr Zinchenko in the summer.
Tierney has started only six times in the Premier League and wants to push for an exit in the summer.
According to several sources, Manchester City and Newcastle United are interested in the Scottish defender.
The 25-year-old defender has three years left on his contract and Arsenal would demand in the north of £30m for his services.
Ex-Scotland player Kenny Miller has urged Tierney to make a move to Eddie Howe’s side before the new season commences.
“I think he probably needs to now. When he plays, he’s a world class left-back. To play regular football, he probably needs to step away from Arsenal,” Miller said.
“There will be lots and lots of teams that will take it. He’ll have loads of potential suitors. If he moves, he’ll have a number of options.
“He’s one of our best players. A move to Newcastle with everything that’s going on at that place at the moment would be a really big step. Champions League football as well, so he’s not going to be stepping away from that level.”
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