The Portuguese international was a rumoured transfer target for La Blaugrana last summer before opting to remain with the Premier League champions in 2021/22.
City boss Pep Guardiola hinted at a willingness to sell the former AS Monaco schemer if he is not happy with his squad role at the Etihad Stadium.
However, as Silva is under contract at City until 2025, they could demand around €80m to release him.
Any possible interest from the Catalan giants depends on whether Frenkie de Jong joins Manchester United, with his sale providing the main funding for Silva.
According to reports from Mundo Deportivo, the Catalan giants are monitoring the situation and could make a bid before the end of the month.