But the 21-year-old Durham seamer's father, Alan, who suffers from a hereditary kidney disorder, turned him down, believing it would adversely affect his cricketing hopes.
The former chemical technician from Middlesbrough, north-east England, said the offer came after his health worsened and doctors said he needed either renal dialysis or a transplant.
Neither of his two brothers or sister were compatible as they all had the same condition, polycystic kidney disease.
"The only person in my family who could give me a kidney is Liam," the 49-year-old was quoted as saying by the weekly Sunday Sun regional newspaper in north-east England.
"As soon as he found out he offered me one straight away without any consideration whatsoever about his career. All he was worried about was me and my health and he simply said, 'take one of mine'.
"You could not do anything more for a parent but I said, 'thanks but no thanks'. It was an amazing thing to do but I could not accept because I'd be worried it would affect his cricket career.
"It could have set him back a long way but he was not concerned about that at all."
Liam Plunkett is part of England's 15-man squad for the World Cup after playing a key role in the visiting side's victory over Australia in the Commonwealth Bank one-day series in February.
His father managed to see England's one-day triumph Down Under but is unable to travel to the West Indies.
The cricketer said of his offer Sunday: "With close family on their death bed you would, wouldn't you?
"He said he wouldn't take it because of the career I have ahead of me. I was fully prepared but he wouldn't accept it anyway.''
Alan Plunkett underwent a liver transplant 10 years ago and may require a second transplant in the future. He is now facing lengthy dialysis sessions three times a week until a suitable donor is found.