Sir Danvers Osborn was the son of Sir John Osborne and his wife, the Hon. Sarah Byng, the only surviving daughter of Admiral Sir George Byng, later Viscount Torrington. He succeded his grandfather, Sir John Osborne, 2nd Baronet to the baronetcy in 1740. He married, in September 1740, Lady Mary Montagu, daughter of the 1st Earl of Halifax in the second creation (see lots 138 and 143), who died giving birth to their second child, John. In 1742 he became MP for Bedfordshire and in 1745, during the Rebellion of the the Young Pretender, he raised a troop of men and led them in person to support the king. In 1750 he went to Nova Scotia for six months to visit the governor, Lord Cornwallis, and three years later he was appointed Governor of New York Province. He sailed from Portsmouth on 22 August to assume his new office and arrived on 6 October. Six days later he died. He never recovered from the loss of his wife and it was speculated that he may have taken his own life. Danvers, a town in Massachussets, is named after him.
It was he who changed the spelling of the family's surname from 'Osborne' to 'Osborn', in order to avoid confusion with the family of the Duke of Leeds, their cousins.