"This is the defeat that hurts the most in my five years here, there’s no doubt about it," he said after a result that leaves the U.S. without a point from two matches in the Hexagonal qualifying group.
"It is a bitter pill to swallow."
Speculation over Klinsmann's tenure as head coach was already rampant before the game, even as U.S. Soccer president Sunil Gulati remained publicly supportive.
After the loss to Costa Rica, Gulati called for a metaphorical time-out, as reports linked former head coach Bruce Arena to the job.
"We won't make any decisions right after games," Gulati said. "We'll think about what happened today and talk with Jurgen and look at the situation."
If Klinsmann is to be dumped, now would appear the perfect time to do it, with four months until the next qualifier.
The U.S. still have a good chance of qualifying for Russia 2018, and extending their streak of appearances in World Cup Finals to eight, but Tuesday's result, on top of the home loss against Mexico on Thursday, leaves little margin for error.
Perhaps more troubling, some observers questioned whether the players had lost faith in Klinsmann and his system.
The top three teams after the 10 home-and-away matches from the relatively weak CONCACAF Hexagonal qualify for Russia, with the fourth going into a playoff against a team from Asia.
Costa Rica head the table with six points, followed by Mexico and Panama with four, and then Honduras on three. The U.S. and Trinidad and Tobago are pointless.
Klinsmann, 52, a former champion striker, has been in charge of the U.S. men's team since 2011.
He acknowledged that his future would be a talking point.
"I understand that when you lose two games and especially two World Cup qualifiers right after each other, that there will be a lot of comments," he told reporters.
(Reporting by Andrew Both in Cary, North Carolina; editing by Amlan Chakraborty)
