The Olympic long jumper's partner, Susie Verrill, revealed Rex's plight after initially believing he was suffering side effects from a routine vaccination.
But after the couple's one-year-old tot took a turn for the worst, former Strictly Come Dancing star Greg, 31, and 30-year-old Susie were told to take Rex to hospital.
He was then "blue-lighted" to another hospital for further tests, with doctors fearing he was suffering from a bowel condition.
Alongside a photograph of Rex strapped to a stretcher in the back of an ambulance, worried Susie told her Instagram followers: "We got blue-lighted to a 2nd hospital earlier – they thought Rex might have a weird (but kind of common) bowel thing going on & needed another hospital to do scans."
She later shared another picture of Rex lying in bed asleep, with a drip attached to his hand.
Susie added: "Good news – it wasn't the bowel thingy.
"Bad news – they don't know what it is but think it might be a bad virus.
"Hoped he was coming out the end of it but he just woke me up vomiting everywhere again so now we're waiting for the cleaners.
"He won't sleep in the cot so he's on the hospital sofa bed with me."
Former magazine journalist Susie, has since confirmed that doctors have diagnosed Rex with both anaemia and a "virus".
After spending the night in hospital with their son, Susie said on Tuesday morning: "We are being allowed to go home today, we think it's just a virus but he's also anaemic."
She added: "It's not the cause for the vomiting but it's not going to be making him feel very good.
"Hopefully we can get a bit more food on board."
Greg and Susie, who met in 2012, recently celebrated Rex's first birthday.
They are also parents to son Milo, who was born in November 2014.
The couple first revealed their youngest boy was under the weather on the weekend, telling fans they believed he was suffering from a bad reaction to a routine Meningitis B vaccination.
Alongside a heartbreaking picture of Rex, Susie said: "I rang the doctors for advice yesterday (after another round of vomming) and was told to bring him in.
"Turns out it probably is indeed a bad reaction to the Men B vaccination, combined with the joy of teething.
"He’s had a slightly better day, but is NOW battling constipation. Being a baby, not easy is it?"
But posting a video on Monday, she revealed: "We are in the hospital. Rex is still poorly.
"We had to take him to the doctors this morning because we were meant to take a urine sample in to check he didn't have a UTI, but he hasn't weed enough over the past four days to even get a sample.
"He's still being sick so we are now in paediatrics."
She later admitted she had "ummed and ahhed" about speaking so publicly about Rex's illness, but added: "I can feel a bit iffy chatting about the less glossy stuff sometimes."
Susie has thanked hers and Greg's followers for all their kind messages.
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