They form part of the historic Bridgewater collection owned by the duke and on display at the National Gallery of Scotland.
The collection also includes three paintings by Raphael, a Rembrandt, a Van Dyck, a Tintoretto and two more Titians.
'Most important'
The 7th Duke of Sutherland has decided to sell some of the collection to raise £100m.
His offer comes with the added incentive that, should the two galleries manage to raise the necessary funds, the entire collection will remain on long-term loan to the Scottish gallery.
"The Bridgewater Loan ... is the most important Old Master paintings loan to any public museum in the world," said the gallery's director, John Leighton.
Losing Diana and Actaeon, he added, "would be like the Mona Lisa being taking out of the Louvre".
Advertisement
A look at the paintings in the National Gallery of Scotland
Speaking to the BBC's Today programme, Mr Leighton maintained the investment was essential, despite current pressure on public spending.
"We would never argue that expenditure should be diverted from all those important areas - whether it's health, defence, or whatever - but we do have to keep spending on culture," he said.
"Every so often you just have to make a major investment in the quality of our National Collections.
"Not every year, not every ten years, [but] once or twice a century and that moment is now."
Under the deal, the galleries will be offered each painting in turn. They have until the end of this year to commit to buying the first.
Titian (Tiziano Vecellio) is acknowledged as one of the greatest colourists of all time, who pioneered the use of oil paints, and paved the way for generations of artists to come.
Bookmark with:
What are these?