The host nation fell at the quarter-final hurdle, giving up a 2-1 lead to 10-man Japan who went on to win 3-2 with a last-gasp goal.
Qatar lost their opening match to Uzbekistan with Metsu saying at the time that he was grateful not be fired. But they bounced back to beat China and Kuwait.
Qatar Football Association President Sheikh Hamad bin Khalifa bin Ahmed al Thani said they would discuss the situation this week, but suggested Metsu's job could be safe.
"We wanted to show Asia and the world that we have a good team and we succeeded in that," he said.
"I am happy with the performance of the entire team and the coach. They gave their best.
"The QFA will convene a meeting next week to evaluate the team’s performance in the tournament. There we will talk with Metsu.
"We will discuss with him the best course for Qatar football. With World Cup qualifiers not to start before June next, we have plenty of time and will not take a hasty decision."
However another QFA official, quoted by the Qatar Tribune, said it would be tough for Metsu, as he had plenty of time to prepare the team for the Asian Cup and could have done better.
"It failed in two Gulf Cups, World Cup qualifiers and the Asian Cup," he said.