With his hand strengthened by the £13.7m Juventus paid for Patrick Vieira, David Dein, the Arsenal vice-president, is understood to have offered Seville a deal in excess of the club record £17.5m Arsenal agreed to pay the same club for Jose Antonio Reyes.
Jose Maria Del Nido, the Seville president, said: "We have not concluded an agreement between the clubs but Arsenal are getting close to the financial figure we could accept. They have more than doubled the £8.3m they started with and it would surpass the fee we got for the sale of Reyes. But obviously the agreement is between three parties."
It is a significant "but". After two seasons in Spain, Baptista can begin the process of taking out dual nationality but would not be able to complete it until well into 2006. That status would save him the rigours of applying for a work permit when moving countries and, equally significantly, would make him an EU player under Spanish regulations. As Baptista himself has said: "If I would go to Arsenal or another foreign team, I wouldn't get a Brazilian-Spanish passport. After a third year at Seville, or another Spanish team, this won't be a problem."
Arsenal fear it has been suggested to him that if he waits until then, Barcelona (who have already filled the three-man quota of non-EU players allowed in La Liga) will sign him.
"We are in no-man's land at the moment," said the Arsenal manager, Arsène Wenger. "I am very keen to get him. He has great strength and can play up front or midfield. He is versatile and can score goals. But there are a lot of ingredients in there.
"The situation is quite positive with the club but we have not sorted it out with the player. It is 50-50. It is down to him. He might stay where he is or go to another club in Spain."
Tottenham have also expressed an interest in Baptista but that is unlikely to worry Wenger. Barcelona may be more alluring to a Brazilian than Arsenal but Tottenham are not.
Baptista joined Seville for less than £2m just two summers ago. However, having been signed as a midfielder he was converted to a striker, scoring 20 goals in his first season in La Liga, helping Seville to qualify for the Uefa Cup. His uncompromising style of play earned him the nickname "The Beast".
The second season can often be the most difficult one but Baptista scored another 18 goals in 2004-05 as Seville again qualified for Europe. Baptista, despite the strong competition, then won a place in Brazil's squad for the Confederations Cup, making two brief appearances as a substitute.
While Wenger is open about his desire to sign Baptista, he was dismissive about reports linking him to Edgar Davids and Olivier Dacourt. In the wake of Vieira's departure, Arsenal seem to lack experience in central midfield but Wenger said: "I'm happy with what we have. I will add what I feel is really needed. Numbers-wise we don't need to add players but if we find a special player we'll take him."
On Dacourt specifically, Wenger said: "No, there is no truth in that at the moment." On Davids he added: "Again there is no real truth behind that. There are a lot of players on the market. We have a big squad and we'll look for maybe just one addition."
Ideally, that would be Baptista.