The Catalan giants are aiming to freshen up their attack ahead of next season by bringing in Antoine Griezmann from Atletico Madrid in a £100m deal.
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Barca needs to free up some funds to help push through the transfer, according to Spanish outlet Don Balon, which could mean cashing in on one of their prolific marksmen.
Suarez netted 82 goals in 133 appearances for the Reds before leaving for Camp Nou in 2014 and has since gone on to net 110 times in La Liga for Barcelona.
The report claims that Liverpool would be at the front of the queue if the Uruguayan striker is made available to purchase, although competition could be provided by Inter Milan.