The 29-year-old midfielder, whose contract expires at the end of the season, has netted 11 goals this season and has admitted he is in the form of his life.
Such form has reportedly attracted interest from Hibernian but Wright will try to persuade the former Dundee United, Peterborough and Coventry player that his future is best served by staying at McDiarmid Park.
Wright, who lost out to Hearts in a previous attempt to sign Swanson, said: “I informed the club last April that I wanted Danny to be re-signed. Unfortunately it hasn’t happened yet.
“I don’t want him to leave, he’s our best player. I feel as though I get the best out of him. He’s had a torrid time at Hearts. Down in England it wasn’t great for him. His best spells of football have been with me and St Johnstone.
“So hopefully we are in the box seat and that should help us get him to re-sign, but whether he does or not, I don’t know.
“I can’t think of too many that left here and went on to do better. Sometimes a player can find a club that he is comfortable with and hopefully that’s the case, but I don’t know if he will sign or not, that’s the bottom line.”
Wright is unaware of any interest from Hibs, the club Swanson supported as a boy.
“I know Neil Lennon well,” the former Northern Ireland goalkeeper said. “I would imagine if there was any interest he would pick up the phone. I know he will do things the right way.
“It’s getting to that time of the season, press are going to speculate and print things that may or may not be true. I couldn’t tell you if it’s true or false, we will have to wait and see.
“They are quite within their rights to show interest if that’s the case.”
Saints host Inverness on Saturday and Wright expects to have Joe Shaughnessy, Zander Clark and Murray Davidson in his squad following injuries.