Grenville was the grandson of Sir Bevil Grenville, and the son of Bernard Grenville, M.P., and groom of the bedchamber to Charles II, by his wife Anne, daughter and sole heiress of Cuthbert Morley of Hornby, Yorkshire.
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Grenville was born near Withiel, west of Bodmin, Cornwall, the son of Sir Bernard Grenville by his wife Elizabeth Bevil, and was a grandson of Sir Richard Grenville (1542–1591), the heroic Elizabethan naval captain, explorer, and soldier.