The board decided to keep faith in Lopetegui until Monday's Premier League clash with Wolves, the BBC reported on Thursday, after 14th-placed West Ham went down 3-1 at Leicester City earlier in the week.
"I have big commitment to my work, always, and I am very happy to stay at West Ham. We are going to change the situation," said Lopetegui, who took over in May.
"I understand all the things around football and I understand my main aim and responsibility is to be ready for a tough, hard match.
"But in the same way, it's a big opportunity and challenge for us at home, to be ready to achieve the three points. That is the most important thing."
Former Spain and Real Madrid coach Lopetegui faced chants of "you're getting sacked in the morning" from West Ham fans after they suffered their seventh loss in 14 league games at Leicester, placing them six points above the relegation zone.
"I accept the responsibility and the pressure. That's why I am a coach," Lopetegui said.
"For sure, we're not happy and, for sure, the fans are always right... they give us much more than we give them.
"We have to change this, and we are looking for this."