Abidal, 34, who now plays for Monaco, was diagnosed with a liver tumour and underwent an operation in March 2011.
He had a transplant in April 2012 with part of an organ donated by his cousin.
The story with Dani goes beyond the jersey. He knows it. Me too. We are friends and we talk about everything. He wanted to support me
"When I had to undergo surgery, Dani wanted to donate his liver," said Abidal in an interview with Spanish radio station Catalunya.
"Logically, he couldn't because he is a professional footballer."
Abidal made a playing comeback in April 2013 after his liver transplant, but was released by Spanish champions Barcelona in the summer.
Brazil right-back Alves, who was Abidal's team-mate at the Nou Camp for five seasons, decided to wear the France international's old number 22 shirt following his return to Monaco.
Abidal added: "The story with Dani goes beyond the jersey. He knows it. Me too. We are friends and we talk about everything. He wanted to support me.
"And besides, he's a very good person. He was my neighbour and his ex-wife and my wife are friends."
Abidal also tweeted: "Thanks brother" alongside a picture of himself and Alves playing for Barcelona.
Ex-Barcelona defender Eric Abidal has revealed former team-mate Dani Alves offered him part of his liver when the Frenchman needed a transplant.