The Tykes are braced for further offers for the 26-year-old, who they signed from Everton on a permanent basis in January 2008, with fellow Championship side Preston also reported to be interested in the midfielder.
Barnsley manager Simon Davey told the club's official website: "We received a bid for Anderson from Bristol City last month, but we turned it down.
"We have a valuation for every player and the bid received was nowhere near that.
"Everybody has a price. Just look at what has happened with Ronaldo and Kaka in the last week, but Anderson is a very, very important player to Barnsley Football Club and our valuation reflects this."
De Silva arrived at Oakwell for an initial loan spell in November 2007 before Everton agreed to make the move permanent for an undisclosed sum.