Los Che were bundled out by Morocco in the round of 16 at the World Cup on Tuesday, with Spain unable to convert any of their spotkicks as the North African side won 3-0.
Spain have lost four of their five penalty shoot-outs at the World Cup, which is the worst record of any nation.
It also comes after losing to Russia on penalties at the same stage four years ago, while they went down 4-2 in a shootout to Italy in last year’s Euro 2020 semi-finals.
Enrique had taken over after the 2018 World Cup but the Spain coach was unwilling to discuss his future after Tuesday’s defeat, with his contract expiring at the end of the World Cup.
“I cannot tell you, I don’t know this decision,” Enrique told reportrers.
“This is not the right time to discuss my future, it’s not relevant.
“My contract is going to end, but I am very happy with the national team, with the federation, and I have always had great support from ([Jose Francisco) Molina, who is the sporting director.
“I could always carry on. I need to have peace of mind to decide what’s best for me and the team.”
Spain midfielder Sergio Busquets, who turned 34 earlier this year, also declined to talk about his future with the national team.
“Now the important thing is the team and not me,” the Barcelona midfielder said.
“It’s a difficult night and we will have to get up and use this experience, there are very young people who will be very useful and we must continue.”