Spain produced a lethargic display in its last-16 clash with Italy at the Stade de France, going down 2-0 to goals from Giorgio Chiellini and Graziano Pelle, prompting questions over Del Bosque's future.
And while the 65-year-old insisted there would be no snap decision, he conceded it may be taken out of his hands.
"For my personal situation, I have to speak to the president and we'll think about it then," he said.
"He's the one who brought me here and we'll have a conversation. He will do the best for Spanish football."
The defeat ended Spain's dominance of European football following their trophy successes in 2008 and 2012.
Del Bosque was adamant, however, it was not the end of an era.
"No I don't think so. Spanish football has a great system, we've been working for many years," he said.
"There are good youth academies and good players. The work of the clubs is good so this era will not finish
"Maybe we will understand how difficult it is to win major championships."
Del Bosque did admit that his side was second best on the day to an Italy outfit whose vibrancy his players could not match.
"I think in the first half we were a bit timid. We didn't play with enough bravery or decisiveness as we did in the other games," he said.
"In the second half we took a lot of risks and we tried to get the equaliser. We tried but we couldn't.
"Italy were better and I wish them all the best.
"We had good intentions. You can't doubt our good intentions.
"To say they battered us is excessive.
"In sport, sometimes the opponent is better, but it isn't the case that we haven't played with enough hunger."
Del Bosque named the same starting line-up for the fourth successive game, but rubbished suggestions his players were fatigued.
"I don't think we were tired," he said. "We tried until the end, keeping Italy in their own half. Okay, there were some counter attacks but I don't think it was about tiredness.
"Our team is not a physical team, we are a technical team."