Abidal, 38, who is now the club’s technical secretary, had a transplant in 2012 after a tumour was found in his liver.
Reports in Spain claim Rosell, who was arrested for money-laundering in 2017, purchased the organ.
Barcelona say they “roundly deny any irregularity in the matter”.
The claim has also been denied by Abidal and the Hospital Clinic de Barcelona, where the player had the operation.
Barcelona added: “The club are saddened by the lack of rigor in the spreading of such information about such a sensitive issue.”
Abidal, who spent six seasons at the Nou Camp, made his comeback for Barcelona in April 2013 before joining Monaco three months later.
Last week a Spanish court ruled that Rosell, who is in custody, will stand trial on charges of money laundering involving the sale of television rights for Brazil matches.