Chris Gard and Connie Yates, who are in their 30s and come from Bedfont, west London, want a doctor in the USA to treat their baby son Charlie Gard.
Specialists at Great Ormond Street Hospital in London, where Charlie is being cared for, say therapy proposed by a doctor in America is experimental and will not help.
They say life support treatment should stop.
A High Court judge in April ruled against a trip to America and in favour of Great Ormond Street doctors.
Mr Justice Francis concluded life support treatment should end and said Charlie, who suffers from a rare genetic condition and has brain damage, should be allowed to die with dignity.
Three Court of Appeal judges upheld that ruling in May.
A panel of three Supreme Court justices - headed by Lady Hale, the Deputy President of the Supreme Court and most senior female judge in British legal history - is scheduled to consider the couple’s claims at hearing in London on Thursday.
Supreme Court officials say justices will consider preliminary arguments before deciding whether to give permission for a full appeal hearing.
A spokesman said if permission was granted, a date for a full hearing would be set.
Doctors are continuing to provide treatment pending decisions by Supreme Court justices.
Mr Justice Francis had made a ruling on April 11 after a trial in the Family Division of the High Court in London.
He heard that Charlie, who was born on August 4 last year, had a form of mitochondrial disease, a condition which causes progressive muscle weakness and brain damage.
Specialists in the USA have offered a therapy called nucleoside.
Charlie’s parents have appealed for money on a GoFundMe page to cover doctors’ bills in America.
They reached a £1.2 million target before the High Court trial.
More than 83,000 people have donated money and the fund has now topped £1.3 million.
Mr Justice Francis said Great Ormond Street doctors had considered the experimental treatment, but decided it would not help Charlie.
He said the case had never been ‘’about money’’.