The 24-year-old World Cup winner is continually linked with a move back to his boyhood club during every transfer window.
Last summer, the Gunners rejected a reported £30 million (Dh176.1m) offer for their inspirational captain.
However, with Fabregas appearing more determined to rejoin the club he left as a teenager in 2003, Arsenal are said to be ready to allow the deal to go through, if Barcelona improve their latest £27m bid.
BBC Sport quote a senior Arsenal official as saying: "The offer was made formally in writing to our chief executive [Ivan Gazidis], and we said no straight away."
When asked if the club expected an improved offer this week, the source added: "Possibly yes, and if it's enough then I expect we'll have to sell."
Last week, Sandro Rosell, the Barcelona president claimed the European champions would not break the bank for Fabregas, of whom he said: "it is clear that this year his value is less."
During the 2010 summer transfer window, Arsenal released a statement confirming Fabregas was not for sale. However, so far, there has been no official comment from the Premier League club following renewed speculation over his future.
The sale of Fabregas would allow Arsenal to press ahead with a summer recruitment spree.
The London club been linked with Gervinho, the Ivory Coast forward, but have yet to make an offer, according to Michel Seydoux, the Lille president.
The 24 year old, who won a domestic double with Lille last season, has made no secret the Gunners would be his preferred destination were the French club ready to sell.
Seydoux told Radio Monte Carlo: "We know that Gervinho is keen on Arsenal. For the moment, there has been simple contact, but we have not received a concrete offer yet."
Arsenal have also been linked with Lille's Belgian starlet Eden Hazard, but Seydoux maintained the 20-year-old midfielder would remain at their club.
It has so far been something of a slow transfer window for the Gunners, with Carl Jenkinson so far the only confirmed addition in a £1m deal to take the young defender from Charlton Athletic.
Arsene Wenger, the Arsenal manager, is under pressure to bolster his squad to help end a six-season trophy drought.
The addition of Southampton's 17-year-old forward Alex Oxlade-Chamberlain is expected to be confirmed at the start of July ahead of Arsenal's Far East tour.
However, the quest for more experienced cover, particularly in defence, continues to prove a source of frustration.
Having sold Phil Jones to Manchester United for £16.5m, Blackburn Rovers are now in a position to hold out for an increased fee for Christopher Samba, his fellow central defender.
Spain's Under 21 midfielder Juan Mata is another reported target, rated at around £18m by Valencia, while the Gunners are also said to be tracking the Argentine youngster Ricky Alvarez.
Wenger is also waiting on Samir Nasri and Gael Clichy to sign new contracts.