The France international underwent a lengthy operation on Tuesday which finished at about midnight, just after Barca's 4-0 La Liga victory at home to Getafe during which the whole stadium chanted Abidal's name and applauded the fullback.
"Today could be a great one all round, because we know that Abidal and his cousin (Gerard, the donor) are recovering well after the operation," Barca president Sandro Rosell said on Wednesday.
"I can't say anything more than this because the family have specifically asked us not to say any more."
Abidal's Barca team mate, Spain midfielder Sergio Busquets, wished him all the best when he spoke at a news conference on Wednesday.
"The operation is over and the next few hours are very important," he said. "In two or three days we'll see how it is all going.
"From here I went to send him all the very best, as everyone else has been doing."
Abidal had surgery to remove a tumour from his liver last March before making a triumphant and emotional return to lift the Champions League trophy at Wembley just two months later.
Last month, the club said Abidal would need a transplant but did not say how long he would be out of action.