 He was the fifth son of Sir George Bourchier (d. 1605), an English soldier who settled in Ireland, the 3rd son of John Bourchier, 2nd Earl of Bath (1499–1561) of Tawstock, Devon, by his 2nd wife Eleanor Manners, daughter of George Manners, 11th Baron de Ros.Henry's mother was Martha Howard (c. 1555–1598), daughter of William Howard, 1st Baron Howard of Effingham.Thus his uncle was Charles Howard, 1st Earl of Nottingham, Lord High Admiral.