Any deal is likely to be put on hold till after the tournament if no agreement can be found by Wednesday, when the 23-year-old should be given the day off on the eve of travelling to South Africa with the Spain squad.
Barca have indicated they are ready to play a waiting game till the last day of the transfer window in their pursuit of Fabregas.
But there is a nagging desire within the Nou Camp for outgoing president Joan Laporta to complete a deal before elections to replace him take place on Sunday.
Reports in Catalonia say Barca technical secretary Txiki Begiristain and director of football Raul Sanllehi are ready to fly into London for face-to-face negotiations in which they will present an improvement on their £29million opening offer of last week.
Arsenal insist their captain is not for sale at any price, though privately it is believed they are now looking for about £60m.
If that valuation cannot be chipped away at, a deal is unlikely this week, as the Spanish champions are reluctant to go above £40m. They are prepared to offer players in part-exchange to break the impasse.