City chiefs have been angered by reports the club have already tabled an offer to Messi ahead of a possible free transfer move to the Etihad Stadium in the summer.
Messi, who has won the Ballon d'Or six times, is free to leave Camp Nou at the end of the season as his contract expires and City, as well as Paris Saint-Germain, are among the clubs interested.
Messi has been free to talk to clubs outside Spain since January but sources have told ESPN he has not yet made a decision on where he will play next season and his representatives have not held talks over potential moves.
Sources have told ESPN that City will wait to see what Messi decides before making any potential move at the end of the season.
It is the same stance they employed in the summer when the 33-year-old looked set to leave after submitting a transfer request, before agreeing to stay with the Spanish giants for another year. City insist no offer was made in the summer before his decision was announced.
Barcelona, meanwhile, have seen their finances hit hard by the coronavirus pandemic while there is continued uncertainty at Camp Nou because of the ongoing presidential elections. A new president is set to be elected on March 7 following the resignation of Josep Maria Bartomeu in October.
Barcelona's financial problems, plus a potential early exit from the Champions League following a heavy 4-1 defeat to PSG on Tuesday, have fuelled speculation that Messi is set to end his association with the club in the summer.
Joan Laporta, Victor Font and Toni Freixa are the three candidates in the running to replace Bartomeu, with each saying publicly they want to keep Messi and ensure he ends his career at Barcelona, where he has scored 654 goals in 760 games.
Suarez, 34, joined Atletico last September in a deal reported at the time to be worth just a few million in performance-related add-ons.
However, La Liga leaders Atletico actually agreed to an initial fee of €5m for the striker, who is the top scorer in the Spanish league this season with 16 goals, and a further €6m in variables.
Atletico have already paid €2m of those variables because Suarez has made 20 appearances and they will have to pay €2m more if they reach the Champions League quarterfinals by beating Chelsea. The final add-on, also worth €2m, is related to performances next season.
The Rojiblancos did not have to pay the initial €5m out of their own pocket, though. The money was knocked off a debt Barca have with them relating to the signing of Antoine Griezmann in 2019.
Barca paid Griezmann's €120m buyout clause but also paid an additional €15m -- officially as a first option on several Atletico players -- when Atletico threatened to take them to court in search of €80m more because talks with the French striker began while his clause was still €200m.
That €15m is being paid in instalments, but €5m was knocked off when Suarez made the move to the Wanda Metropolitano.
Barca are also topping up Suarez's salary until the end of the season, when his contract would have expired at Camp Nou. He earned around €20m annually with Barca, but Atletico are paying less, so the Catalan club are making up the difference.
Information from ESPN's Barcelona correspondents Moises Llorens and Sam Marsden was used in this report