The report claims the Champions League holders are ready to strengthen their ranks during the next transfer window, despite the fact they have already secured the signatures of Arda Turan and Aleix Vidal from Atletico Madrid and Sevilla respectively.
Such a move for Dutchman Van Persie would come as little surprise, though, given that Lionel Messi's long-term injury and the sale of Pedro in the summer has left Barca short of attacking options.
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The Spanish media claim that Enrique holds Van Persie's experience and ability in high regard, and considers him the perfect backup to main striker Luis Suarez.
Whilst luring the clinical striker to the Camp Nou wouldn't be an issue, though, it is hard to see current club Fenerbahce giving up on their star man, who joined them in the summer.
And in an attempt to bolster their defensive options also, it would seem that the La Liga champions are in pole position to land PSG centre-back Marquinhos.
The Brazilian arrived in France in 2013 for a hefty €31 million from AS Roma but failed to establish himself as a first-team regular, due to competition from the likes of David Luiz and Thiago Silva in the heart of PSG's defence.
Enrique will be fully aware that Barca's backline could benefit from some fresh blood, with centre-backs Javier Mascherano and Jeremy Mathieu on the wrong side of their 30's.
Barcelona currently sit top of La Liga, one point ahead of rivals Real Madrid, but have already shown signs of struggling without talisman Messi given their narrow 2-1 victories over Las Palma and Bayer Leverkusen, and their away loss to Sevilla in the league two weeks ago.