The Mirror said talks were held last week amid Messi’s “growing dissatisfaction with the state of the Barcelona squad.
It said Messi was serious considering leaving the Calatan giants following Neymar’s world-record departure to Paris Saint-Germain.
Although Messi agreed terms on a new contract earlier in the summer, the deal remains unsigned.
The Mirror said Manchester City’s interest is long-standing and they are understood to be prepared to use his £275 million release clause to sign him if the possibility presents itself
The paper quoted Barcelona sources as saying Messi “is seriously considering leaving.”