The Norway international has enjoyed a superb season with the Gunners, playing a pivotal role in their Premier League title charge that ended in a second place finish behind champions Manchester City.
PSG are expected to make major changes to their squad this summer with the futures of both Lionel Messi and Neymar uncertain.
According to The Mail, Odegaard is one of the players the French giants are keen to bring in as part of efforts to reshape the squad after a disappointing campaign
Their efforts to prise him away from north London are unlikely to prove unsuccessful, however, with the 24-year-old happy in north London.
Gabriel Martinelli and Bukayo Saka have both signed new contracts at Arsenal with the club also in talks with William Saliba, who has one year left on his current deal.
The midfielder initially joined the Gunners on a six-month loan from Real Madrid just over two years ago, before completing a £35million transfer the following summer.
Odegaard was handed the armband on a permanent basis during the close season following the departure of Alexandre Lacazette.
Mikel Arteta is keen to make changes to his squad after they fell short in their title bid with a number of Arsenal stars likely to be moved on.
Odegaard is not among them, however, viewed as a vital to their plans to overtake City next season.
Arsenal will have a transfer budget of around £180million this summer that can also be topped up through player sales.
Granit Xhaka is likely to leave after scoring twice in what may have been his last game for the club against Wolves on Sunday, while Thomas Partey is another Arteta is open to selling. Kieran Tierney and Rob Holding are also among those who could be moved on.