The United States international's move from the Major League Soccer outfit is subject to him passing a medical and obtaining a work permit.
Shea was a spectator at Stoke's 2-2 home Barclays Premier League draw with Wigan on Tuesday night and after the game, Potters boss Tony Pulis spoke about his optimism of wrapping up a deal for the 22-year-old.
"I think it is close now, and we are hoping that we can get that deal done in the next day or so," said Pulis on the club's official website.
"Brek is a very, very exciting player, and one who we believe could go on to become a very good player for the football club.
"He has terrific pace, a very good left foot and he is capable of scoring goals.
"If you look at the clippings of him on the internet then you will see what he is all about.
"I believe he has all the tools needed to be a success here, so hopefully we can get the deal done."
The development comes less than a week after Pulis said the move was dead in the water.
"We enquired about whether we could bring him over, not managed to achieve it but we move on," he said at the time. "That didn't happen, draw a line under it."