The 29-year-old is locked in negotiations with the La Liga side over an extension to his current deal - due to expire at the end of next season - with wages thought to be a significant stumbling block.
The Argentina international currently earns £500,000 a week at Nou Camp and is pushing for an increase following an £850,000-a-week offer from the Chinese Super League, but Barca are unwilling to commit to significantly extravagant sums, having already locked down Neymar and Luis Suarez to new big-money deals.
According to The Sun, City held informal talks with the La Liga side late last month to see whether a deal for Messi would be possible.
The newspaper claims that City did not make a formal offer for Messi at the meeting but "made it clear that they were not afraid" to beat the world-record fee of £89m that rivals Manchester United paid for Paul Pogba last summer.
The talks were described as "positive" by one source, with City thought to be planning to offer Messi a significant wage boost to £800,000 a week before tax.
Messi has spent his entire career with the Catalan giants, scoring 326 goals in 363 La Liga appearances to date.