roving eye

noun2

1. •
   Wide-ranging observation of one's surroundings.

   • Mr Codlin had relaxed into a grim smile as his roving eye detected hands going into waistcoat pockets and groping secretly for sixpences. — 1840
2. •
   The personal characteristic of taking amorous interest in people other than one's own spouse or regular romantic partner.

   • This dame she was new-fangled And of a roving eye. . . . "Beshrew me," quoth King Arthur, "I think thou be'st not true!" — 1858