The 22-year-old is undergoing a medical with the Potters after arriving in England shortly before the 2-2 draw with Wigan at the Britannia Stadium on Tuesday.
Shea flew to England after Stoke reached an agreement with FC Dallas and the MLS, although the deal is subject to him passing a medical and obtaining a work permit before the close of the transfer window on Thursday evening.
Pulis said: "I think it is close now, and we are hoping that we can get that deal done in the next day or so.
"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."