Health chiefs have said a temporary closure of some or all paediatric services at Pilgrim Hospital, Boston, may be needed in order to maintain patient safety.
Bosses at United Lincolnshire Hospitals NHS Trust (ULHT) said an inability to recruit enough properly trained staff means closing children's services from June 4 must be among one of four options considered.
The trust said keeping things as they are is "reliant on finding additional multi-professional staff from agency to cover children's, maternity and neonatal services", and "getting the right balance" of agency and full-time staff.
A petition to save the hospital's children's ward has more than 5,000 signatures, while a public meeting to discuss the issues is due to be held this week.
Critics have claimed the loss of dedicated children's in-patient, emergency paediatric and some neonatal care "could put lives of children at risk".
Sharon Poulter, who set up the petition, said: "As the largest county in England, with it's population spread over a large area of mostly agricultural land, it is imperative we keep this vital children's ward open."
Speaking on the petition website, she added: "Where are all the children going to go? This could put lives of children at risk."
Health campaigners fear patients could instead be diverted to Lincoln's County Hospital, or to A&Es or wards in Nottingham, Peterborough and Leicester.
The trust board has asked for an all-options report for May 25, when it takes a final decision.
Raising the matter in the House of Commons, Conservative MP Matt Warman asked the Prime Minister to "leave no stone unturned" to "make sure that we recruit the doctors we need".
Theresa May, speaking on May 2, replied that the trust "wants to continue to provide paediatric services at Boston, and every effort will be made to ensure that that can continue".
In August 2016, ULHT closed the county's Grantham and District Hospital's overnight A&E services as a "temporary" measure following staff shortages, after a safety review recommendation.
More than a year after the decision was taken, the emergency department is still on daytime-only opening.
Campaign group Fighting 4 Life Lincolnshire has pointed to that measure claiming it is evidence any temporary closure could effectively become permanent.
It has said the loss of "vital services" in paediatrics could leave the county "fragile and unsafe".
In a comment on its Facebook page, it said: "This is about life and death and the future of our services."
Dr Neill Hepburn, ULHT medical director, said the decision was not about saving money but patient safety, adding: "All options being discussed will cost more money than now."
He said: "A temporary decision will be made for one reason and one reason only and that is to ensure our children's services remain safe, as in July we forecast that we will only have one substantive middle grade doctor working at Pilgrim.
"This isn't about saving money, it's about safety.
"All options being discussed will cost more money than now.
"We have had a long-standing shortage of children's doctors and nurses at ULHT and have carried out extensive worldwide recruitment."
He added: "These efforts haven't resulted in the numbers of doctors needed to maintain current services."
Elsewhere, Worcestershire Acute NHS Hospitals Trust closed children's accident and emergency services at the Alexandra Hospital, Redditch, in September 2016, as a "temporary" measure, citing staff shortages.
Emergency under-16 cases are now taken to the nearest alternative paediatric A&E at Worcestershire Royal Hospital in Worcester, up to half an hour away, or to Birmingham.