A retracing of his transfer steps has been mooted for the Brazil international.
He left Camp Nou for France as part of a record-breaking €222 million (£200m/$262m) deal during the summer of 2017.
Goals and trophies have been delivered at PSG, but Neymar is said to be looking for a way out of Parc des Princes.
It has been suggested that Barcelona would be willing to welcome the South American back onto their books, but no transfer talks have been opened by the Catalan club.
Cardoner told reporters when quizzed on the links to Neymar: “It is not the time to talk about arrivals.
“We are in a process of departures; it is a period that expires on the 30th of this month.
“The correct thing is to say that Neymar wants to return to Barcelona, but I do not agree that Barcelona worry about signing Neymar, this is not on the table.
“There are many things about his departure that I did not like, many things.
“There are claims that we have signed Neymar. We have not signed anyone. We have not kept in touch with him. He and many more want to play for Barcelona.
“It does not surprise me that Neymar wants to come back, it has already happened before with others and sometimes they have returned – Cesc [Fabregas], [Gerard] Pique.
“Neymar is a great player, but the circumstances of how he left also have to be evaluated to see if this can become a reality.”
A move for Atletico Madrid striker Antoine Griezmann has been mooted for some time.
He has opened the door for a transfer this summer, a year on from opting to close it, but the France international is another that the Barca board are yet to approach.
Cardoner adder: “I understand that any player has the goal of coming to Barcelona, where the best players in history have been.
“All players can be willing to come and want to play, but we have to have a balanced budget.”