Trundle has been one of the most prolific strikers in League One in recent seasons and has been linked with a move to a higher level.
Robins boss Gary Johnson is looking to strengthen his attacking options after securing promotion to the Championship and is looking at a number of players.
Trundle is understood to be at the top of his list of targets and Swansea have been asked about his availability.
Reports suggest Bristol City have already had more than one bid for Trundle rejected, but chairman Steve Lansdown insists negotiations are still at an early stage.
"We are looking at several different forwards and Lee Trundle is a player we recognise," he told the Bristol Evening Post.
"We have had a conversation with Swansea, but there is nothing further to report at this moment in time."
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