The Scottish left-back, 22, joined the Gunners in a deal worth £25m (€27m/$30m) on transfer deadline day and will compete with Sead Kolasinac and Nacho Monreal for a starting spot once he recovers from his hernia operation.
And 34-year-old midfielder Brown feels that, should the young defender excel, he could earn himself another big move.
“You don’t know what could happen,” Brown told The Herald.
“He could have a fantastic couple of seasons and end up at Barcelona or PSG or wherever.”
He also insisted that there are no ill feelings from the Celtic squad towards Tierney for moving on and claimed the club would welcome him back one day if he chose to return.
“There are pictures on the wall all over the club of players who have left, and everyone would wish them all the best. They can come back as well, that’s what this club is all about, he said.
“I see Ajax bringing players through and selling them on - and they have a party for them leaving. We try and criticise. It’s a different level.
“We wish him all the best. We just concentrate on now and [new left back Boli Bolingoli] has to push on and try to get to the levels that Kieran did.”
Bolingoli signed as Tierney’s replacement from Rapid Vienna and started the season as Celtic’s first choice left-back while the former’s future was resolved.
He made 42 appearances across all competitions in Austria last term, scoring two goals and recording five assists.
Those numbers mirror Tierney’s attacking output almost exactly, with the Scotsman managing one goal and six assists from 40 appearances in all competitions last season.
The last Celtic defender to make a high-profile move to the Premier League was Liverpool’s Virgil van Dijk, who then joined Southampton, and Tierney will hope he can be as successful as the Dutchman has been in his time in England.