The British heavyweight surrendered his unbeaten professional record in a surprise second-round knockout defeat to the American in Liverpool on Saturday.
A return bout is now on Price's agenda and the 29-year-old has said that he would be happy for Thompson to have home advantage if it would give him the opportunity to avenge his shock loss.
"If it meant me taking a little bit less money to get the chance to avenge it then it is something we could look at," Price told BBC Sport.
"Going to America to fight would all be part of the learning experience. We will talk about it this week and if we could get him [Thompson] to agree to do that, that would be great."
Price had been touted to fight compatriot Tyson Fury in an all-English clash before his defeat at the Echo Arena.