Ancelotti has returned to the Santiago Bernabeu to replace Zinedine Zidane, leaving Everton to undertake a second spell at a club where he previously spent two years in charge.
Bale was in the squad that won the Champions League under the Italian in 2014, scoring Madrid's second goal as they defeated Atletico Madrid 4-1 in the final to secure 'La Decima'.
However, the forward fell out of favour under Zidane and spent the 2020-21 season back on loan at Tottenham, scoring 16 goals in 34 appearances in all competitions.
It remains unclear if he has a future with Los Blancos, though, even with a familiar face now back in charge.
"He's returned to Real Madrid and he's a great guy," Bale said about Ancelotti in an interview with Sky Sports.
"We had a great time together at Real Madrid and I'm sure he's going to be amazing in charge there.