City have already turned down a €10m bid for the 19-year-old and want at least €18m for him.
Barca will submit a second offer of €15m but that will be as high as they are willing to go since Garcia is out of contract next year.
Garcia has already said yes to a return to the club he left to join City in 2017.
However, the two clubs are still a distance apart and negotiations could go down to the wire.
If City refuse their next proposal, Barcelona will wait until next summer when they can take the centre-back for free.
City paid €1.5m for Garcia and now stand to make a big profit after he broke into Pep Guardiola’s first team.
Garcia, who made his Spain debut against Ukraine earlier this month, appeared 20 times for City last season.