The France international was diagnosed with a tumour on his liver in March 2011 and, though that was successfully removed by doctors, the Frenchman required a transplant a year later.
And he has now explained that Alves, who wears Abidal's shirt number for Barcelona this season following his friend's close-season departure, was prepared to make a sizeable sacrifice upon hearing the news.
"My relationship with Dani goes much further than him taking my old No. 22 shirt," he told Radio Catalunya.
"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."
Abidal, who made a full recovery from his transplant, left Barcelona upon the expiry of his contract in July in order to return to his homeland with newly promoted Monaco.
He has started all of Monaco's first five Ligue 1 games this term, and has helped his new club to top spot.
