The former Club America tactician took charge in October and, after steering Mexico to a 9-3 intercontinental play-off success over New Zealand, was given the job on a full-time basis.
Mexico, who endured a tumultuous qualifying bid before Herrera's arrival, caught the eye in Brazil and cruised through Group A unbeaten after beating both Cameroon and Croatia, while they also drew 1-1 with the hosts.
But the old last-16 curse struck again as the Netherlands scored two late goals to inflict a sixth successive second-round loss on Mexico, ensuring they fell short of the target set by the Mexican Football Federation.
Upon the team's return home, Herrera and the Federation directors refused to acknowledge whether the 46-year-old will remain as coach, but Dos Santos and Layun are certain he is the right man for the job.
"Since Miguel arrived, he has returned hope to the players and fans," Dos Santos is quoted as saying by El Universal.
"He is a coach who has given so much to the team. Today the team has a very good image.
"On the field, if we haven't been better then we have at least been on par with the great powers, so I am sure Miguel has all the confidence of the Federation and he is a coach will give the Mexican people a lot of joy."
Layun added: "I am sure he will be retained. I've always said, not just here with the national team but also with my club, that continuity is important. Long-term projects are better.
"The coaching staff in general has shown great capacity. (We've shown that) we can put together a competitive team with a winning mentality, which is the most important lesson from this.
"This team never gave up and it’s logical that we continue the (current) process."
