The Brazil superstar become football's most expensive player when he signed for PSG in a €222 million (£194m/$261m) deal in August.
But while Neymar's goals fired PSG to a domestic treble and he won the Ligue 1 Player of the Year award, despite missing the last two months with injury, speculation about his future in the French capital has grown.
Real Madrid have been strongly linked with a move for Neymar, head coach Zinedine Zidane saying he believes the forward would be compatible with Cristiano Ronaldo.
Speaking to El Chiringuito, Bordas backed up comments from former Brazil striker Ronaldo, who said he does not know if Neymar is happy at PSG.
"Neymar can sign for anyone," the Barca director said. "For me, personally, I'm not dreaming of Neymar signing for Real Madrid.
"It seems as if he's not happy in Paris, we figured as much.
"In Barcelona he was doing well and Neymar regrets having left [for PSG]."
Barcelona have been strongly rumoured to be lining up a move for France forward Antoine Griezmann, who scored a double to help Atletico Madrid ease past Marseille in the Europa League final.
"I saw the Europa League final," Bordas added. "He's really good, he's a great player. I congratulate Atletico for their Europa League success."