The 29-year-old's future at the Bernabeu has been in doubt for a number of months amid injury issues and restricted first-team opportunities.
Following Real's Champions League final victory over Liverpool in May, Bale admitted that he was open to leaving the club in search of more game time.
However, The Telegraph has claimed that Bale had face-to-face discussions with Lopetegui on Monday and was told that he has a huge part to play in a new era for the club following the sale of Cristiano Ronaldo.
Lopetegui is understood to have told Bale how much he admires him, and how he is looking forward to working with the Wales international in the forthcoming campaign.
Manchester United have reportedly been monitoring Bale's situation at the Bernabeu ahead of a prospective move for the former Tottenham Hotspur star.