The USA international is currently undergoing a medical at the Britannia Stadium after the Potters reached an agreement with the MLS outfit.
The deal is subject to the 22-year-old obtaining a work permit before Thursday's transfer deadline, but Pulis is confident that Shea will be a Stoke player come Friday.
"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. "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."
Shea has 15 caps for the USA national team.