The Icelandic attacking midfielder has emerged on the radar of Premier League champions Leicester City but Taylor has urged him to remain at the Liberty Stadium as the Swans get set for a big 2016/17 season, provided they keep their star players,
"We are desperate for Gylfi to stay. Once you do well you are going to get linked with the top teams and Leicester happened to win it," said Wales international Taylor.
"We want all our best players to stay and add to that.
"We don't want to be a selling club. The Premier League means you don't have to do that any more, with the money that has come in.
"That is why we've seen clubs in strong positions, and we have seen the league we have this year.
"Hopefully we can hang on to Gylfi and a few others."
Sigurdsson scored 11 goals and provided three assists in the Premier League for Swansea in the recently-completed season.