Abidal's cousin Gerard provided part of his liver for the Monaco player's operation in April 2012 after he suffered hepatitis.
But Abidal, 34, told a Spanish radio station that Alves also said he would provide the organ needed for the transplant.
Alves is wearing Abidal's number 22 shirt this season at Barca as tribute to the player, who had to undergo liver surgery to remove a tumour in March 2011 before returning two months late to play in the club's Champions League final win over Manchester United.
"My relationship with Dani goes much further than him taking my old No. 22 shirt," said Abidal, who move to Monaco during the close season after Barca opted against extending his contract.
"When I had to undergo surgery, he offered to donate his liver. Of course, it’s completely out of the question because he is a professional footballer. Our relationship goes much deeper than friendship."