The Museum's Open Weekends will begin on Saturday 5 June and run on the first full weekend of each month for the rest of 2021.
The limited reopening has been made possible by emergency funding from the Government's Culture Recovery Fund.
A highlight of the open days will be the Museum's special exhibition, Nightingale in 200 Objects, People & Places, which originally opened in March 2020, just 10 days before England entered its first period of lockdown.
The exhibition has, therefore, remained largely unseen and behind closed doors. Created to mark the 200th anniversary of Florence Nightingale's birth in 2020, it presents 200 items, stories and places that explore the reasons for Florence Nightingale's international fame and enduring influence.
"We are extremely pleased to be opening our doors again, if only for a limited time," said museum director David Green.
"These will be very special, now-rare, opportunities to explore the life of the woman whose name has been used so much over the past year and whose legacy shines through the remarkable work of the health care professionals that have been fighting the pandemic.
"Last year, our big plans for Florence's bicentenary were stopped in an instant; we hope you will join us as we begin our comeback."
Tickets for the open weekends are available at www.florence-nightingale.co.uk