Those reports also suggest that Arsenal are yet to receive an improved offer from the £27 million Barcelona offered last month.
It is claimed that Arsenal are prepared to sell the 24-year-old midfielder if the La Liga champions meet their £40 million valuation.
Fabregas is scheduled to take part in Arsenal's final preseason friendly against Benfica on Saturday night, and if he is fit and a deal hasn't been reached, he will be expected to begin the season for Arsenal.
Arsenal will play Italian side Udinese in the playoffs to get into the main Champions League group draw. Having finished fourth in the Barclays Premier League last season, automatic qualification for the tournament was forfeited. The playoff is scheduled for the week of August 16-17 with Arsenal playing at home in the first leg.
This poses an interesting scenario for Barcelona. If Fabregas were to take part in the Champions qualifiers, which it would be assumed he would, it would disqualify him from playing for Barcelona in the Champions League if a deal for the Arsenal skipper were to be reached.
While both sides remain steadfast in their valuation of the player, Arsenal expect Fabregas to continue playing and training with the squad.
Should Fabregas leave, it is thought that Arsene Wenger would revive his interest in Valencia's Juan Mata.