Chief operating officer Clare Louise Smith said Leeds Teaching Hospitals NHS Trust would "like to encourage people to consider other options such as their GP, pharmacy or the NHS 111 service when possible".
The appeal comes amid claims of huge queues at the accident and emergency department at the LGI.
Leeds resident Lauren Middleton, 26, took her five-year-old daughter Ruby to the A&E department on Sunday.
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She had called 111 and said she was advised to go straight to A&E because her daughter was "at risk of a seizure" with a temperature of almost 40 degrees.
However, Lauren claimed that she found queues stretching outside the building when she arrived.
She said: "We were told it was up to two hours to check in and seven hours to see a doctor.
"They were queuing outside lined up in the white tent and into the ambulance car park. That was to just get checked in."
Lauren ended up taking Ruby home under the advice of plenty of fluids and paracetamol before she returned to her GP on Monday.
Ms Smith said the Trust had "robust plans" in place to respond if and when the pressure increases on the services, with a focus on "prioritising patients in terms of clinical urgency".
Addressing the concerns over waiting times, she said: “LGI is seeing significant demand for urgent care like many other hospitals around the country. Patient safety is our first priority and we have robust plans in place to respond if and when pressure increases on our services, prioritising patients in terms of clinical urgency.
"It’s important local people know they will get safe, high quality care if they do need to attend A&E."
Ms Smith said hospital staff were working "extremely hard, under sustained pressure" to keep people safe.
"I’d like to ask people to treat those staff with kindness and respect," she said. "I want to emphasise how proud I am of the whole team – from the clinicians in our A&E department to all the support services – for how they are responding to this demand.
"They work to see patients quickly, whilst working within national guidelines around Covid-19 infection prevention and control, which limit the numbers we can see safely in the emergency department at any one time.
“We’d like to encourage people to consider other options such as their GP, pharmacy or the NHS 111 service when possible.”
As reported in the YEP last week, the number of patients in Leeds waiting over four hours to be seen in A&E has risen to a record high as the city’s hospitals battle “significant” demand for emergency care.
Figures show in August, just 68.7 per cent of A&E patients were admitted, transferred or discharged within the four hours - an emergency care standard - far below the 95 per cent national target and the lowest level seen since at least April 2018.
This is against a backdrop of high levels of attendances at the city’s emergency departments.
Data shows the number of patients arriving at Leeds General Infirmary’s A&E was 17.3 per cent higher in August 2021 than it was, pre-Covid, in August 2019.
Leeds has two urgent care centres which can be accessed by patients who need to see a doctor or nurse urgently, but do not have a life-threatening injury or condition.
The St George’s Urgent Treatment Centre in St George's Road, Middleton, and Wharfedale Urgent Treatment Centre at Wharfedale Hospital in Newall Carr Road, Otley, are both open daily, 8am-11pm.
They can deal with issues such as sprains and strains, simple broken bones, wounds and bites, burns and scalds, and minor head, eye and back injuries. Patients are advised to call NHS 111 to arrange an appointment before attending.
Meanwhile, the Shakespeare Medical Centre at 1 Cromwell Mount, Burmantofts, is open daily, 8am-8pm.
It can be used by none-emergency patients who have a minor illness such as coughs, chest infections, fevers, sore throats and headaches.
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