An initial offer of £250k for the England U-21 midfielder was instantly rejected.
Leeds have confirmed making a second, improved bid believed to be over £500k.
But Burnley are determined to keep the 19-year-old at Turf Moor, where he enjoyed a fine campaign last season.
Sam Ellis, former assistant manager at Burnley, today confirmed: "Yes, we have made a bid for Richard, but that is as much as we are saying. We do our business in private and try to be discreet as we can. We don't like going public on these things."