L'Equipe says PSG could be willing to sell the Brazilian, although not many sides can afford his wages.
Eddie Howe's men are among the few teams that would be willing to pay his salary, though a lack of Champions League football will work against them.
Juventus are also in the running, with boss Massimiliano Allegri keen.
Neymar penned a PSG extension in spring last year, with an extra year set to trigger on July 1.
This will take the deal up to 2027.
Speaking to Le Parisien yesterday, PSG president Nasser Al-Khelaifi refused to commit to a future with Neymar.
He said: "I expected all of the players to do more than last season. Much more!
"The message is clear: working each day at 200 per cent. To give everything we have for this shirt, to give it our all, and then we'll see what the result is.
"We have to become humble again. We have to change to avoid injuries, suspensions and fouls which can change games."
Asked specifically about Neymar by Marca, he said: "Some will come and others will go, but these are private negotiations."
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