Ghana international Ayew joined Swansea from Marseille and scored 12 Premier League goals in his first season, helping the club finish 12th in the table.
France-born Ayew has been linked with possible moves to top-tier sides Sunderland or West Ham United.
'I hope he stays. I don't know, but we have discovered other good characteristics with him,' Guidolin said.
'He says he is not a striker and he is a midfielder but in my opinion, he is a striker. He is also a good false nine.
'He is a big player, but his attitude is good. He is an important player, but he is quiet, a good guy.'