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The Reds talisman handed in a transfer request in the summer but the Merseyside club stayed strong and rejected three bids from the La Liga giants.
Spanish outlet Don Balon report Coutinho has an agreement in place with Liverpool which will allow him to leave at the end of the season.
But the Brazilian is keen to push that forward and wants a move to Barca in January.
Coutinho does not want to risk the Catalan club changing their mind and seeking an alternative target to him to improve their midfield.
Ernesto Valverde is looking for a replacement for Neymar after he shocked the football world by leaving Barca for Paris St-Germain in the summer.
Liverpool WAGs: Match the player to their partner

Coutinho has been in fine form for Liverpool this season, netting nine times and providing seven assists in all competitions.
And the reports claim the 25-year-old is desperate to move as soon as possible and play with the likes of Lionel Messi at Barcelona.
Liverpool are set to demand over £130m for their superstar, a figure which the La Liga leaders will be reluctant to meet.
Coutinho will look to impress for the Reds again when they travel to Bournemouth in the Premier League on Sunday.
The Reds are pushing for a place in the top four but any hopes of challenging for the title are over as they trail Manchester City by 18 points.
Barcelona will look to maintain their La Liga lead when they take on Deportivo on Sunday.