The Brazil international arrived at training on Wednesday with his father and representative, and told the Spanish club he wanted to leave.
He was then given permission by Barca manager Ernesto Valverde not to train and to "sort out his future".
PSG is understood to be ready to pay the 25-year-old's release clause.
The latest development came two days after it emerged Barca was ready to push for a Financial Fair Play investigation if PSG signed Neymar.
That came after La Liga president Javier Tebas threatened, in an interview with Mundo Deportivo, a legal response against the former Ligue 1 champion if European football's governing body failed to take action.
He also said PSG president Nasser Al-Khelaifi had been made aware of the Spanish league's intentions.
Neymar moved to Barcelona from Brazilian club Santos in 2013 for £48.6 million, and signed a new five-year deal with the 24-time Spanish champion in 2016.
His representative Wagner Ribeiro, who on Wednesday tweeted he was in Paris, said last year that his client had been offered a tax-free £650,000 a week to sign for the French club.
He also said the forward "was close to a move to Manchester United".
Rumors of Neymar's potential departure resurfaced recently, but club president Josep Maria Bartomeu told the BBC the player would be staying.
Neymar played for the Catalan side in the recent International Champions Cup in the US, before travelling to China to fulfil commercial commitments.
He traveled back to Spain on Tuesday and returned to the club's training ground on Wednesday.
The Brazilian has scored 105 goals for Barcelona, helping the Spanish giant to two league titles, three Copa del Rey trophies and a Champions League crown.