The midfielder has informed Wenger that he wants a move back to Spain to join his boyhood club, and announced he was resolved to settle his future before Spain's campaign at the World Cup begins next week.
Reports in Spain today claim an increasingly frustrated Fabregas even called Wenger yesterday to reiterate his desire to leave.
Tug of war: Fabregas is wanted by Barcelona but Arsenal are refusing to sell
Barcelona had an opening formal bid of £29million rejected and attempts to keep negotiations open have been difficult. They will try to contact Arsenal today but know that the Gunners value their skipper at about £60m and show no sign of budging.
Focused: Fabregas is preparing to play in the World Cup with tournament-favourites Spain
'What is important is that Cesc has a great World Cup and what is also important is that he comes back to Arsenal and has a great desire to do well for the club,' said Wenger.
'We want to keep our best players, that is my huge determination. I have built this team for years, from 16-17 years of age with most of the players, and we want to continue to work with them because we feel they are very close and the team has many strengths in many ways.'
Barca this week floated the idea of offering Yaya Toure and about £10m, but although Wenger is keen on the Ivory Coast midfielder, he would prefer to keep the deals separate.
The manager's stance follows on from a strongly-worded statement issued by Arsenal last week.The intense speculation surrounding Fabregas's future prompted the Gunners to insist they have 'no intention' of selling the 23-year-old midfielder.
The statement, following the opening bid on June 1, read: 'Arsenal captain Cesc Fabregas is under contract with the club until the summer of 2015. He is a highly valued member of the team and part of our future plans.
Busy summer: Arsene Wenger is also in South Africa for the World Cup, and met localchildren on a visit to the Nike community training centre in Soweto
'We received an offer from Barcelona for Cesc and in response, we immediately and resolutely told them once again that we have no intention of selling our captain.
'To be clear, we will not make any kind of counter-proposal or enter into any discussion. Barcelona have publicly stated that they will respect our position and we expect that they will keep their word.'
Barca president Joan Laporta revealed his frustration earlier this week by saying he didn't understand why Arsenal were trying so hard to keep hold of a player who wanted to leave.
The Catalans have missed the optimum deadline for signing Fabregas before the Spain squad arrive in South Africa for the World Cup on Friday and they are now said to be seriously considering Valencia's David Silva as an alternative.
Talkative: Barcelona midfielder Xavi trains with the Spain World Cup squad
They may have to decide quickly, however, as Manchester City are already more advanced in talks to sign 25-year-old Silva.
Barcelona's shameless public pursuit continued on Wednesday when midfielder Xavi used the announcement of his new contract at the Nou Camp to push home his desire to team up with Fabregas.
'He wants to come here,' said the 30-year-old. 'I always said that hehas Barca DNA and I'm confident there will soon be a solution.
'He must play at the club, and it must be now. If not now, it will bea little later, but I hope that Cesc come to Barcelona as soon as possible.'
Meanwhile, Barcelona presidential candidate Marc Ingla has revealed he dreams of unveiling Fabregas as his first new signing if he wins Sunday's election.'Cesc is a marvel, as is (David) Villa,' he said. 'My only worry is paying over the odds, but I confess that my dream would be to unveil Cesc as president.'
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