The Nou Camp club are thought to be keen to strengthen their defence, with Javier Mascherano and Alex Song having deputised in recent weeks.
The report claims Barca have been tipped off to Kompany's unhappiness at the decision to dispense with Roberto Mancini, and are prepared to test the water with a bid for the defender.
They were linked with Paris St Germain star Thiago Silva, but the Ligue 1 champions made it clear they would not sell the Brazilian international under any circumstances.
With the managerial change at City and Kompany's loyalty to Mancini, the situation is less clear.
Manuel Pellegrini is the man tipped to replace the former-City boss at the Etihad Stadium, and recent reports have linked the Malaga manager with a move for Real Madrid defender Pepe.
Still, the Daily Mirror report claims City will fight hard to stave off interest in Kompany, who captained the club to its first ever Premier League title a season ago.
Barcelona will test the water with a £20m offer for the 27-year-old. Kompany joined City from Bundesliga side Hamburg for a fee in the region of £6m in the summer of 2008, and he's gone on to become one of the Premier League's most consistent centre-backs.