New coach Xavi Hernandez wants to add another attacking player to his squad in January, with ESPN reporting last week that Barca have also sounded out City's Raheem Sterling and Chelsea duo Timo Werner and Hakim Ziyech.
Director of football Mateu Alemany and CEO Ferran Reverter were in Manchester on Friday for a meeting with City to discuss a transfer.
Sources have told ESPN that City's initial response was that they are reluctant to let go of Torres, who is expected to return from an injury at the end of the year.
Torres, 21, is understood to be open to a return to Spain, though, and sources added City could open the door to an exit should he express a desire to leave.
However, they would demand over €40 million for a player they initially signed for a fee of €23m from Valencia in 2020.
Barca's financial situation is likely to hamper them in any negotiations. The club's gross debt stands at over €1 billion and Alemany recently said they are not in a position to register new players in January while adhering to LaLiga's salary cap.
The Catalan club continue to work on reducing their wage bill, with Lionel Messi moving to Paris Saint-Germain on a free transfer while Antoine Griezmann rejoined Atletico Madrid on loan in the summer. Barca are also looking to bring in new sponsors to give them room to manoeuvre moving forward.
Sources added that reaching an agreement with City will still be difficult, though, even if Barca do find a way to amortise any potential fee over the duration of Torres' contract.
A move for Torres has been in the works behind the scenes for several months, sources have told ESPN. Barca first sounded out a move in the summer.
Alemany previously worked for Valencia, where he built a good relationship with Torres and his agents, laying the groundwork for this week's talks.
New Barca coach Xavi has backed the move. He is keen to add another forward to the squad in January and Torres, who can play wide or through the middle, is seen as the perfect fit.
Torres has scored 16 goals in 43 appearances for City and another 12 in just 22 caps for Spain.