Aguirre, who had already made clear he would leave his role after the tournament, said former Mexico captain Marquez was ready to continue the work done with a squad that restored belief during the country's deepest FIFA World Cup run in 40 years.

England booked its place in the quarter-finals after surviving a late Mexico push at Azteca Stadium, where the home side fell short despite a spirited fightback.

"I would have liked to say goodbye to my people with a victory. That hurts," Aguirre said. "We tried."

Jude Bellingham's quick-fire double put England in control before Julian Quinones pulled one back for Mexico. Harry Kane added a second-half penalty after England had been reduced to 10 men, and Raul Jimenez's late spot-kick was not enough to prevent Mexico's exit.

"The players need to leave with their heads held high," Aguirre said. "Today it could not be. We failed to capitalise, but I want to thank them all."

The match, delayed by one hour because of adverse weather around the Azteca, was played in front of a passionate home crowd that Aguirre has often described as Mexico's "12th player".

The fans roared Mexico on in the closing stages, chanting "Yes, we can" as the co-host pushed for an equaliser, but England held firm to advance to a quarter-final against Norway in Miami on Sunday (AEST).

Aguirre said he was leaving with pride despite the pain of elimination, pointing to the unity of the squad and the renewed connection with supporters.

"I leave with a lot of pride," he said. "These players made me very happy. We recovered the sense of belonging and the identity of this team."

Marquez, who captained Mexico at five FIFA World Cups and worked alongside Aguirre during the tournament, is expected to take the project forward after the veteran coach's departure.

"I wish him the best," Aguirre said. "He is more than capable, and he will do better than me."