The Japanese midfielder joined the Etihad Stadium from Gamba Osaka this summer but could seal a loan switch to Tynecastle.
The report also claims that Meshino was also in attendance at Tynecastle at the weekend to watch Hearts' draw with Ross County.
Talks with the attacking midfielder will reportedly continue over the next few days as Craig Levein looks to secure the services of the Etihad summer signing.
Meshino joined Man City for a reported £900,000 (Transfermarkt) from Gamba Osaka this summer, where he rose up the ranks to make the first-team breakthrough and went on to make 12 appearances in the Japanese top flight for them.
He featured a total of 25 times in all competitions for the club last season - in the J1 League, J3 League, J League Cup and Emperor's Cup - scoring 13 goals and registering two assists, according to Transfermarkt.
If the rumours were true, Meshino would become the seventh Hearts signing of the summer transfer window after the arrivals of Craig Halkett, Jamie Walker, Conor Washington, Steven Naismith, Lic Damour and Joel Pereira at Tynecastle.
Hearts have won the Scottish title four times, the Scottish Cup eight times, plus four Scottish League Cups (Transfermarkt).
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