The UK Paediatric Intensive Care Society (PICS) tweeted an alert it said was from NHS England that warned of a "rise in the number of children [...] presenting with a multi-system inflammatory state".
The apparent warning said the condition required "intensive care" treatment.
Professor Russell Viner, president of the Royal College of Paediatrics and Child Health (RCPCH), said parents should be reassured that children are unlikely to be seriously ill with Covid-19.
The tweet from PICS said: "Over the last three weeks, there has been an apparent rise in the number of children of all ages presenting with a multi-system inflammatory state requiring intensive care across London and also in other regions of the UK".
According to the alert, which has also been shared with GPs, children affected display signs similar to toxic shock syndrome (TSS), a severe illness associated with infections, and have blood markers in line with severe Covid-19 in children.
They may also have abdominal pain and symptoms of inflammation around the heart
The alert continues: "There is a growing concern that a SARS CoV-2 (Covid-19) related inflammatory syndrome is emerging in children in the UK, or that there may be another, as yet unidentified, infectious pathogen associated with these cases."
The message sent to doctors talks about atypical Kawasaki disease - a condition that mainly affects children under the age of five.
Symptoms include a high temperature that lasts for five days or more, often with a rash and/or swollen glands in the neck.
ITV News Health Correspondent Emily Morgan explains the new condition
Professor Simon Kenny, the NHS's national clinical director for children and young people, said: "Thankfully Kawasaki-like diseases are very rare, as currently are serious complications in children related to Covid-19, but it is important that clinicians are made aware of any potential emerging links so that they are able to give children and young people the right care fast.
"The advice to parents remains the same: If you are worried about your child for whatever reason, contact NHS 111 or your family doctor for urgent advice, or 999 in an emergency, and if a professional tells you to go to hospital, please go to hospital."
Prof Viner said: "We already know that a very small number of children can become severely ill with Covid-19 but this is very rare - evidence from throughout the world shows us that children appear to be the part of the population least affected by this infection.
"New diseases may present in ways that surprise us, and clinicians need to be made aware of any emerging evidence of particular symptoms or of underlying conditions which could make a patient more vulnerable to the virus.
"However, our advice remains the same: parents should be reassured that children are unlikely to be seriously ill with Covid-19 but if they are concerned about their children's health for any reason, they should seek help from a health professional."
Speaking at Monday's government coronavirus briefing, England's Chief Medical Officer said the possible syndrome among children was "a very rare situation" but added "it is entirely plausible that this is caused by this virus, at least in some cases."
Professor Chris Whitty continued: "We know that in adults who of course have much more disease than children do, big problems are caused by an inflammatory process and this looks rather like an inflammatory process, a rather different one.
"Therefore, given that we have got a new presentation of this at a time with a new disease, the possibility – it is not a definite, we need to look for other causes as well – but the possibility that there is a link is certainly plausible."
One GP, Doctor Natalie Rout, told ITV News she had received the alert via email over the weekend and the update was "worrying".
She added that the public should be made aware of the issue and urged parents to speak to their GP if they are concerned about their child.
Dr Rout added that GPs can assess cases "over the phone and via video conference" if needed.
Professor Martin Marshall, Chair of the Royal College of GPs, said: "GPs are keeping abreast, as far as is possible, of new evidence and information to ensure they can continue to deliver the best possible care to their patients.
"In general practice, at least for the moment, we are encouraged that the cases in children are not increasing, but these particular symptoms sound concerning and we will be working closely with our specialist paediatric colleagues to ensure that we respond appropriately."
He added: "Parents should be reassured that the evidence currently suggests that COVID-19 is generally a mild disease in children, with a low morbidity and mortality rate."
The NHS reiterated that, saying children are contracting coronavirus at the same rate as adults but their symptoms are less severe on the whole.
Coronavirus: Everything you need to know