FAW chief Noel Mooney said manager Page retains the "complete confidence" of the governing body to continue with his current four-year contract.
Page steered Wales to their first finals in 64 years, but they exited at the group stage after a draw with the United States was followed by defeats to Iran and England.
Mooney told BBC Radio Wales Sport: "He (Page) very much has our backing, he has a four-year agreement with us.
"He's a very modern, progressive coach.
"As a manager who's come in and inherited a team, and took it on for a long time as interim manager, took us to a Euros and to a World Cup, we have complete confidence in Rob and he's very reflective."
Page was confirmed as permanent boss earlier this year in succession to Ryan Giggs, having been in caretaker charge since November 2020, and is expected to remain at the helm for Wales' qualifying campaigns for both Euro 2024 and the 2026 World Cup.
Mooney added that an upcoming review of the team's performance in Qatar did not suggest Page's position was in question.
"What you do after a major tournament like this is what you should do, you reflect on that," said Mooney.
"I think we can be confident that the future for Welsh football is very, very good but… all of us need to show a little bit of patience."