Joe-Anne Neeson needs the cash to send three-year-old Caoimhe to the US for cancer treatment that isn't available in the UK.
Caoimhe was diagnosed with a fast-growing, malignant brain tumour three months after becoming unwell.
The 29-year-old mum-of-two from Sandyhills said her daughter was vomiting up to 10 times a day for more than two months, had problems with her balance and agonising headaches before her diagnosis.
She has been told there is only a 40% chance Caoimhe will survive to her fith birthday. It is thought the cancer may also have spread to her spine.
Meanwhile the family is desperately trying to raise the £250,000.
Caoimhe has been turned down for NHS funding for Proton Therapy, a targeted form of radiotherapy which is offered by a number of hospitals in the US.
She is currently being treated with chemotherapy at Glasgow's Sick Kids' Hospital, which has stabilised the cancer but it is probable that it will come back, and more aggressively.
Days after her cancer was diagnosed, Caoimhe was on the operating table.
However, surgeons were unable to remove all of the tumour, because of its size and location.
Joe-Anne said: "It's such a fast growing aggressive tumour. If they had caught it before they might have been able to remove it all.
"Her prognosis is dismal. She has a 40% chance of being alive when she is five.
"She has been amazing through the treatment."
After the surgery Caoimhe, who has an 11-year-old brother Corrie, was left unable to talk and walk but has amazed medics and her family by regaining her mobility and her speech.
Joe-Anne and her best friend Jaclyn Smith, 28, have launched a fundraising campaign to send Caoimhe to the States for Proton Therapy, which delivers a more precise dose of radiation without killing healthy tissue surrounding the tumour. Conventional radiation could leave her severely mentally impaired.
Joe-Anne, said: "The chemotherapy is keeping it at bay but it is very likely it will come back.
"We know she is going to need radiation.
"Proton radiation is targeted – it's like the difference between a shotgun blast or a sniper.
""It all rests on the money. The whole of the East End has rallied behind us. Every day we are getting phone calls from people.
"Hamleys has offered to help and Celtic Park are allowing us to have collecting buckets at the Celtic Barcelona game."
Jaclyn said: "Caoimhe has surprised us all. She couldn't walk or talk after the surgery but is now doing both."
However, Joe-Anne believes that an earlier diagnosis might have helped her daughter. And she has made a complaint to NHS Greater Glasgow and Clyde.
She claims her little girl was originally treated for a bowel obstruction.
Experts say early detection of brain tumours can have a significant bearing on the outcome of treatment.
Joe-Anne, said: "Her whole body was screaming, brain tumour and they failed to diagnose it.
"She presented with every symptom, dizziness, vomiting, balance problems. She was sick every day for 68 days.
"They kept sending me home with her. I looked at her symptoms online and brain tumour kept coming up."
An investigation has been launched by NHS Greater Glasgow and Clyde into Caoimhe's care.
A spokeswoman for NHSGGC said: "We have received a complaint from the patient's mother.
"We are looking into the concerns that she raised with us and will respond to her direct.".
"Decisions to send paediatric patients to America for Proton Therapy are not taken by individual health boards but rather each case is individually assessed by a UK panel of cancer specialists who determine if patients meet the criteria for treatment.