Katie Amos, 30, and fiancé Lee Johnston, 29, from Lincolnshire, were walking through Central Park when she went into labour 11 weeks before her due date.
Their son Dax was born later that day (December 28) weighing just 3lbs.
Katie and Lee had planned to stay in America for just five days, but now face a three month stay as doctors say Dax won't be ready to fly home until his original due date of March 10.
The couple have been told that their medical bills are expected to reach $200,000 (£130,000). They say they are waiting nervously to find if the bill for their baby will be covered by their travel insurance, although they had been told that Katie would be covered to fly abroad.
Lee told the Mirror: "We feel quite stranded at the moment, we are stuck here without our family. It is scary because we don't know anyone so we are completely on our own.
"We just really want to get home but the main thing is Dax is as safe and as well as possible."
Katie and Lee are currently staying in accommodation arranged for them by the Lennox Hill Hospital in Manhattan, where Kate gave birth.
They have set up a fund raising page, with a message written in the voice of baby Dax.
"Hi, my name is Dax Johnston," the message reads. "My Mummy and Daddy went on a 5 day break to New York as a last get away before I came along!
"They went to a cake shop (Mummy loves cakes), and were walking through Central Park on a movie tour and I decided they were having such a good time I wanted to see what all the fun was about, and I decided to pop out 11 weeks early!!!!!
"Oooops! Maybe it wasn't a good idea as now we are not aloud to leave the hospital until March the 10th!!!!!
"We need your help as Mummy and Daddy have no money, no clothes, no friends or family or a familiar face to just say hello to! I know they are scared, but they have some good friends back home that are working very hard to help me! "
So far they have received £6,248 in donations from 691 backers.
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