The two clubs each believe that the other has an advantage when it comes to landing the 21-year-old, who is being chased by Europe's biggest sides.
And the situation is further complicated by a mutual interest in Ajax centre-back Matthijs De Ligt.
Barca spent the summer trying to open negotiations with Ajax for De Jong, with Goal having revealed that former technical secretary Robert Fernandez spoke with Ajax director of football Marc Overmars on numerous occasions , but the €30 million (£26.2m/$34.8m) fee on offer was well below the Dutch club's valuation.
De Jong has spoken of his desire to play with Lionel Messi at Camp Nou on several occasions, and sources at Manchester City had been of the belief that the Primera Division champions were in pole position.
But senior Barca sources now believe that it is City who have the upper hand thanks to a recent push by sporting director Txiki Begiristain, who has travelled to Amsterdam for talks on several occasions.
Barca are resigned to the fact that they would not be able to match City's financial muscle, although Manchester United's Paul Pogba does feature as an alternative midfield option. Paris Saint-Germain's Adrien Rabiot, whose contract is up next summer, is also under consideration.
There is a feeling within the Catalan club that City could even try to buy De Jong in January and loan him back to Ajax until the end of the season.
De Jong, for his part, insists he has no plans to leave the Amsterdam club : "For the moment I'm staying at Ajax, at least another year and maybe two or three more, I'm really enjoying it here," he told Elf Voetbal on Thursday.
Yet that public stance is not enough to put off his suitors.
City are chasing midfield reinforcements after missing out on top target Jorginho to Chelsea in the summer, and the Premier League champions are also drawing up their plans for next summer's window. A January move has not been ruled out; last season they moved to bring in a centre-back and a forward to aid their push for honours.
The versatile De Jong, who can play in several positions in defence and in midfield, is regarded as a special talent by virtually every leading European club and City do not want to miss their opportunity to sign him.
Ajax are hoping to take advantage of the situation to recoup as large a transfer fee as possible.
City have also been working on a deal for Wolves midfielder Ruben Neves , although the Midlands club have indicated that they will fight hard to keep the 21-year-old Portugal international, or at least hold out for as close to £100 million as possible.
It remains to be seen whether City are ready to buy just one midfielder - an effective replacement for Jorginho - or two. Lyon's Tanguy Ndombele is another player that City have been watching closely for several months.
It is believed that City are also ready to move for De Jong's Netherlands and Ajax team-mate De Ligt.
The Blues were heavily linked with a big-money bid for the 19-year-old defender towards the end of last season, but Goal understands that the club were merely laying the groundwork ahead of a move in 2019.
Barca are also interested in De Ligt and it has been reported recently that they will now prioritise a move for him over De Jong, partly because they believe City have stolen a march on De Jong, and because they see De Ligt as the next Gerard Pique.
As with De Jong, however, Barca would not be able to match the English club's financial terms if a formal offer is made.