The 18-year-old, who plays for Major League Soccer club Chicago Fire, is being chased by elite clubs across Europe who are all desperate for his signature.
Despite reports suggesting Wolves are set to sign the goalkeeper for £10million, it is understood the figure is closer to £3million.
The deal is not yet agreed and Wolves have plenty of competition with a number of top clubs interested in signing him, with Real Madrid, Bayern Munich and Chelsea all understood to be interested.
Wolves hope that the pathway they can offer for young players to play first-team football will be attractive to Slonina and will help Wolves secure the signing. The decision now rests with the youngster, who will have the final say in where he moves.
However, as the MLS season does not end until early October, it is not expected that any move for Slonina will be imminent. With the World Cup starting the following month, too, the youngster will be focused on his national team if he gets the call-up.
That could mean that any transfer for Slonina could be delayed until much later in the year or even early 2023.
Slonina made his senior debut for Chicago Fire in August last year, making him the youngest starting goalkeeper in the league's history at just 17 years and 81 days.
He has represented the United States at four different youth levels and has twice been called up to the senior side, but has yet to earn a cap.
Last week he was called up to the senior Poland team for their upcoming Nations League games but he rejected the call-up and pledged his international future to the United States.