As the Gunners have a limited summer transfer budget, it is unlikely that both big money purchases would happen.
Despite the excitement of a possible Zaha signing among Arsenal fans, Seaman believes that adding Tierney to the defence would be the shrewder move.
The former Gunners goalkeeper told football.london: “Yes, Zaha is a good player who could do well for the club, and yes I know I would say as a former goalkeeper that the defence is the first place to look at in terms of signing new players – but the simple fact is that the backline needs strengthening.
“It’s no good having the best players in the world up front if you don’t have a solid defence.
“It’s no secret that Arsenal need to strengthen their defence, certainly after missing out on Champions League qualification. I think Tierney would be an ideal addition for Arsenal.
“He’s young, he’s talented. I think I’m right in saying he’s already captained his club and country and by all accounts he’s a hard worker with a great attitude.
“Being British, he’d also settle in straight away and there would be no period of adjustment. I really think they should be looking to sign Tierney before setting their sights on Zaha.”