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 are understood to be ready to pay the 25-year-old's release clause.
The latest development comes two days after it emerged Barca were 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 champions if European football's governing body failed to take action.
Some of your information will be collected when you use this feature. Find out more
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 £48.6m, and signed a new five-year deal with the 24-time Spanish champions 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".
Rumours 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 United States, before travelling to China to fulfil commercial commitments.
He travelled back to Spain on Tuesday and returned to the club's training ground on Wednesday.
The Brazilian has scored 105 goals for Barcelona, helping them to two league titles, three Copas del Rey and a Champions League crown.
Analysis - 'This transfer has a distinct political overtone'
BBC Radio 5 live sports news correspondent Richard Conway
"Revons plus grand"
PSG's motto is hard to avoid at their stadium, Parc de Princes.
The translation? "Dream bigger"
PSG's owners, Qatari Sports Investment (QSI), are the upstarts within the European game and determined to disrupt the old order.
QSI is an arm of Qatar's sovereign wealth fund - so PSG's ambitions are backed by the enormous natural gas wealth of the Gulf state.
Qatar is locked in a bitter dispute with its Arab neighbours - so this transfer has a distinct political overtone as it seeks to show it will not be diminished or cowed by an ongoing trade blockade and a diplomatic war of words.
Having secured the World Cup in 2022, Doha sees PSG as one of the ways through which the state can achieve its ambitions.
Uefa's Financial Fair Play regulations have certainly tempered PSG's elaborate spending in recent seasons, and Neymar's arrival would represent a serious challenge to the club in this respect - but there's confidence within PSG's boardroom that it will not be a significant issue.
Ultimately this deal will not be judged in the governmental salons of Doha nor on a spreadsheet. Instead, success must be delivered through PSG capturing arguably the club game's greatest trophy, the Champions League.
What will Barcelona be missing?
Neymar is one third of Barcelona's 'MSN' strikeforce, which also includes Lionel Messi and Luis Suarez.
In the past three seasons, they have scored 250 goals between them in 299 appearances.
But they do more than just score...
Read More (...)