Carter Cookson has suffered three cardiac arrests and is on a life-support machine.
Doctors at Newcastle's Freeman Hospital have advised the family that time is running out.
Carter's parents, Sarah and Chris Cookson, are running a social media appeal to find a donor.
The couple, from South Shields, South Tyneside, said they waited five years to try for another child after their first son, Charlie, died aged two.
'Most terrible experience'
Mr Cookson has now said: "Our fear of not taking [Carter] home is becoming a reality now.
"We've got hours, maybe a day, maybe two days."
Mrs Cookson, 44, said in a statement: "We have already lost one son and if we can't find a donor straight away, we now face the prospect of losing our second beautiful baby boy.
"It's just the most terrible experience imaginable.
"I just hope there may be some family out there who can identify with our heartbreak and may be able to help."
A candle-lit vigil was held outside South Shields Town Hall on Thursday evening.
South Shields Labour MP Emma Lewell-Buck last week raised the donor appeal at Prime Minister's Questions.
Theresa May described it as a "tragic case" and highlighted the "opt-out" organ donation system due to be introduced next year.
Mr and Mrs Cookson launched a charity supporting families of children with life-limiting illnesses in 2013 following the death of their two-year-old son, Charlie.
He died of an undiagnosed condition - unrelated to Carter's - which caused problems with his muscles and bones and compromised his immune system.
Family friend Angela Moore said: "Sarah and Chris are suffering so much pain and heartache at the moment as they watch their new born, second son, Carter John, fight for his life.
"Since Charlie gained his angel wings, they have dedicated their lives to raising much needed funds to help families of children with life limiting conditions.
"But now I'm appealing for any other families out there to step forward and help them."https://www.geezgo.com/sps/52367
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