Betis offered little resistance as they were hammered by Los Blancos at home, with fans chanting “go now” at Poyet after the final whistle.
The Uruguayan only joined the Spanish side in May and Betis Vice President Jose Miguel Lopez Catalan says Poyet retains the full backing of the club, stating he needs more time to implement his plans.
“Yes, we fully support the Coach. We’re eight league games in, we have eight points and we haven’t found the game we want yet,” he told Movistar+.
“It’s something that Gustavo also recognises, but we must continue working so we can reach our [desired] results.
“It’s normal for the beticos to be upset, we’re all very upset. The fans didn’t deserve it.
“We’ve built a new team, with 12 signings and a new Coach, and this is something that takes time to bear fruit. We have to keep working.
“On one hand, Real Madrid are a great team, but I consider it more to Betis’ detriment, especially on a defensive level.
“We gave the game away in the first half with a series of mistakes that, at this stage, should not happen.
“After two weeks of the international break, we worked hard, had no injuries and were moving forward.
“But there were then very clear defensive mistakes and we couldn’t get back up.”
Southampton have announced plans for a two-week tour of China as part of their pre-season preparations for the 2018-19 season.