Rayburn said he has talked with "roughly five" other cities about moving the team. He will not identify the cities.
None of the cities has offered any deals or incentives for the Suns to move there, he said.
He would ask of those cities the same request he is making locally: That local government help fund construction of a new stadium.
As is normally the case with minor league baseball teams, the funding for a proposed new stadium would be partial government funding and partial private funding with the owner also making some contribution, he said.
Most new stadiums for the other Class A South Atlantic League teams cost between $10 million and $25 million, he said.
He does not have a particular cost estimate in mind for a new stadium here or elsewhere although it would be less than $20 million, he said. He has also not determined how much money he would want from the city, county and state governments, he said.
If another city offers to build the Suns a new stadium, he would tell Hagerstown officials about that offer before making a decision, Rayburn said.
"If we get a terrific offer, we would run it by the city," he said.
With the Nov. 5 election, the voters elected five Republican candidates, the majority of whom have said they oppose using tax dollars to pay for the stadium.
Rayburn said he is not giving up on getting county financial support for a new stadium and plans to try to take each of the commissioners, particularly the new ones, to Suns games next season.
Newly elected Councilman John C. Munson said he would never support a county-funded stadium.
"I've never been to a ball game in my life," Munson said earlier this month.
"I am taking him to a game and I'm not taking 'no' for an answer," Rayburn said Friday.
If a new stadium is built here, he would prefer it happen at property off Interstate 81 near Salem Avenue because it could draw from a larger area than the current location, he said.
He also is open to the possibility of renovating the existing stadium but he is less enthusiastic about that possibility, he said.
Rayburn owns Big Game Capital, a company based near Cleveland. He also owns the Daytona Cubs, a Class A team in the Florida State League.
He bought the Suns in June 2001 for an undisclosed amount from Winston Blenckstone.