The two attractions have been closed since the end of last year due to the heightened Covid-19 restrictions on mainland Scotland
The two sites, which were completely closed between March an August last year, drew a combined audience of more than 4.4 million when they were fully operational in 2019.
Tickets for timed slots will have to be booked in advance for when they reopen.
The Edinburgh museum was Scotland's busiest visitor attraction in 2019, ahead of London sites like Tate Britain, St Paul’s Cathedral, the National Portrait Gallery and Westminster Abbey.
It will reopen on Monday 26 April, the official reopening date for the Scottish tourism industry under the country’s route map out of lockdown restrictions.
National Museums Scotland, which runs the Chambers Street site, said it would also be reopening the National Museum of Flight, in East Lothian, and the National Museum of Rural Life, in Lanarkshire on the same date.
It has also announced that it will be unveiling its exhibition on the Galloway Hoard, a collection of Viking-age objects found by a metal detectorist in Dumfriesshire, at the end of May.
NMS director Dr Christopher Breward said: “We can’t wait to welcome visitors back once again to all our museums. I know that our many visitors, just like me, will have missed visiting our museums and enjoying the wonderful collections on display.
“We are re-opening once more with our full range of safety measures in place to ensure everyone can have a safe and enjoyable visit.”
Historic Environment Scotland, which runs Edinburgh Castle, said it would reopen on Friday 30 April, along with other major heritage sites, such as Stirling Castle, Fort George in the Highlands, Caerlaverock Castle, in Dumfries and Galloway, Blackness Castle and Linlithgow Palace, in West Lothian, Glasgow Cathedral and Doune Castle, near Stirling.
HES chief executive Alex Paterson said: “Scotland’s heritage attractions are a key part of our tourism sector both nationally and at a local community level, and contribute to our individual well-being, so we’re delighted to once again be opening sites up across the country.
“This will see us reopen all of the sites we reopened last year on the 30 April as well as further sites across the country on a phased based over the upcoming months.
“As always, the safety of our staff and visitors has been at the forefront of our planning, enabling safe access to our properties in line with Scottish Government guidance and our minimum operating standards which underpin our approach.
“We’re also pleased to tell the story of the sites through technology so visitors have the opportunity to experience the old and the new when they are exploring some of our most iconic sites and enjoying a piece of 5,000 years of history.”
Last week the National Trust for Scotland announced that a number of its gardens attractions would be opening to the public on 2 April, with castles, historic houses, museums and visitor centres due to reopen from 30 April if Scotland’s emergence from lockdown is continuing as planned.