The family of nurse Pauline Cafferkey says doctors "missed a big opportunity" to spot she had fallen ill again.
Ms Cafferkey is in an isolation unit in London after tests indicated the Ebola virus is still present in her body.
The health board confirmed she was sent home by an out-of-hours doctor in Glasgow earlier this week.
In an interview with the Sunday Mail newspaper, her sister Toni Cafferkey said it was "absolutely diabolical" the way the nurse had been treated.
Ms Cafferkey, from Cambuslang in South Lanarkshire, spent almost a month in isolation at the beginning of the year after contracting the virus in December 2014.
Bodily tissues can harbour the Ebola infection months after the person appears to have fully recovered.
On Tuesday, the 39-year-old was admitted to the Queen Elizabeth University Hospital in Glasgow after feeling unwell.
She was later flown to the Royal Free Hospital in London where she remains in a serious condition in an isolation unit. She is not thought to be contagious.
Toni Cafferkey told the Sunday Mail that her sister had gone to a GP out-of-hours clinic at the Victoria Hospital in Glasgow on Monday night but the doctor who assessed her diagnosed a virus and sent her home.
She said: "At that point me and my family believe they missed a big opportunity to give the right diagnosis and we feel she was let down. Instead of being taken into hospital, she spent the whole of Tuesday very ill.
"I think it is absolutely diabolical the way she has been treated... We don't know if the delays diagnosing Pauline have had an adverse effect on her health, but we intend to find out.
"It has not been good enough. We think there have been major failings and we just want her to pull through. This kind of recurrence seems to be rare but we don't yet know enough about it."
NHS Greater Glasgow and Clyde confirmed that Ms Cafferkey did attend the New Victoria Hospital GP out-of-hours service on Monday.
A spokesman said: "Her management and the clinical decisions taken based on the symptoms she was displaying at the time were entirely appropriate.
"All appropriate infection control procedures were carried out as part of this episode of care."
On Friday, a statement from the Royal Free Hospital confirmed Ms Cafferkey had been transferred to the hospital "due to an unusual late complication of her previous infection by the Ebola virus".
It added: "The Ebola virus can only be transmitted by direct contact with the blood or bodily fluids of an infected person while they are symptomatic, so the risk to the general public remains low and the NHS has well-established and practised infection control procedures in place."