Williams netted his 17th goal of the campaign as the Glovers condemned Chesterfield to relegation from League One on Saturday and the wheels are in motion to keep hold of the 25-year-old.
"We have already put in our pleas," Johnson told the Western Morning News. "Everybody has an opportunity to sign and, after a decent season, everybody appreciates where they were and what that club has done for them.
"We are not going to be able to entice him through sentiment but I do not mind saying that Andy has probably been made the biggest offer this club has ever made somebody and it is on his table now.
"We made that offer early so it will be up to him. He has obviously got to look at his options, and that's the professional thing to do.
"I for one hope he stays because I think we are going to be competing next year and it will be nice to have him with us and not against us."