Fabregas has long been linked with a return to Barca, the club where he plied his trade as a youngster before moving to the Gunners seven years ago.
The 23-year-old is under contract to Arsenal until the summer of 2015 but last month expressed his desire to head back to the Nou Camp at a promotional event in the Catalan capital.
Barca are keen to make that wish become reality, and on Tuesday submitted a written offer to Arsenal that Laporta has said was worth 35million euros.
That was instantly rejected by the Premier League club, but Laporta is not giving up just yet.
He said in Spanish daily AS: "It's a price that we consider to be his market value. Arsenal have rejected it. Now the sporting director (Txiki Begiristain) must decide what should be the next step.
"We are in negotiations and we'll see how they turn out. At times, these operations are resolved at one speed, and others at another.
"We have to do everything we can to convince them (Arsenal) that the best thing for all is to reach an agreement."
The outgoing Barca president remains optimistic the Primera Division champions will be able to secure the services of the Arsenal skipper, but admits it may not be easy persuading the Gunners to let their prize asset leave.
"The club have full confidence that Arsenal will end up understanding the situation," Laporta said.
Laporta also feels that the ideal situation for Fabregas would be to "have the transfer closed as soon as possible, before the start of the World Cup".
"But it will be difficult because Arsenal are entitled to stand firm and that must be respected."
Laporta also played down suggestions that Fabregas could invoke FIFA's Article 17 rule to force his exit from the Emirates Stadium.
Article 17 regards compensation for breach of contract.
"I'm not sure if Cesc will take this course, but we want to find an agreement by talking with Arsenal," said Laporta.
"We respect Arsenal a lot, we must have in-depth dialogue and follow this path."