Gaggiol claims that the offers were up to €150m (£131m), enough to make Asensio the third most expensive player in the world behind only Neymar and Kylian Mbappe.
But Real, who tied Asensio down to a long-term contract with a €700m buyout clause last September, turned down both bids from the clubs, who Gaggiol stopped short of naming.
"He has an extraordinary future ahead of him."
After breaking into the Real first team last season following a successful loan at Espanyol, Asensio has gone from strength to strength in 2017/18.
He has played in 52 games in all competitions and will be hoping for a role in the Champions League final against Liverpool.
Should Los Blancos beat Liverpool in Kyiv, it will be a record 13th European title for the club, and a fourth in the last five years. Their recent Champions League dominance is such that they could also become the first team in 42 years to win three European Cup titles back-to-back.