It is "not fair and not right" that some parents have to return to work before their newborn leaves hospital, Theresa May has said as she launched a consultation on parental leave.
Under government plans, new parents in Britain would get one week of state-funded leave and pay for every week their baby is in hospital.
The intention is for parents to have more time at home with their newborns.
Every year around 100,000 babies go into neonatal care after their birth.
The consultation will also seek views on how parental leave can be changed to "better reflect our modern society".
Mrs May - who is due to step down as prime minister next week - said she wanted to provide further support for parents dealing with "the unimaginable stress" of their babies being taken into neonatal care.
"Parents have more than enough on their plates without worrying about their parental leave running out and having to return to work before their precious newborn comes home," she said.
"That's not fair and it's not right. So we're also proposing a new neonatal leave and pay entitlement to make this time a bit easier for parents whose babies need to spend a prolonged period in neonatal care."
Concerning parental leave, Mrs May said parenting had changed over the past 40 years "but too often, it is still mothers, not fathers, who shoulder the burden of childcare".
"It is clear that we need to do more and that's why today we have launched a consultation calling for views on how we can improve the current system."
Although the UK's maternity leave provision is above average among leading economies, its paternity leave is six weeks shorter than the average.
The government argues changing paternity leave could promote better gender equality in work and at home.
Women and Equalities Minister Penny Mordaunt said: "Fathers should not have to rely on annual and unpaid leave if they want to be involved in the first months of their child's life."
The consultation will also look at requiring firms to publish their leave pay and flexible working policies.
Paternity rights in the UK
- Mothers are entitled to up to 52 weeks' maternity leave
- They must take at least two weeks' leave after the baby is born (or four weeks if they work in a factory)
- They are eligible to be paid for six weeks at 90% of their average weekly earnings and 33 weeks at £149 per week or 90% of their average weekly earnings (if lower)
- Fathers can take two weeks' statutory paternity leave at £149 a week