A group of senior Wales players, including Gareth Bale, has urged manager Chris Coleman to stay on after his contract ends next summer.
Wales’ hopes of qualifying for the 2018 World Cup in Russia ended with a 1-0 defeat by the Republic of Ireland in Cardiff on Monday.
Coleman, 47, said after the game that he was “not thinking about” his future, having previously said this would be his last campaign.
He was been in charge since 2012.
Wales have not qualified for a World Cup since 1958, although they reached the semi-finals of Euro 2016.
Defender Chris Gunter said: “Everyone in Wales, fans and players, would want to see him stay on, 100%.
“He’s been a massive part of what we’ve done and he is the man to hopefully take us forward.
“If the powers that be can give him a really good contract and make him stay, he can carry on being the greatest Welsh manager of all time.”
Asked about his future, former Fulham boss Coleman said: “I’m genuinely not thinking about it. There’s a dressing room full of devastated players and staff.
“If you look at the make-up of our squad and the age of the squad – some of the young ones coming through – this is a good group of players, even better for tonight.
“Defeat hurts but you learn a lot from defeat – you learn about yourself first and foremost.
“The whole nation will be mourning and disappointed. Again that elusive World Cup has passed us by.”
Wales, who needed to beat the Republic of Ireland to qualify for the play-offs, suffered their first defeat since the 2-0 loss to Portugal in the Euro 2016 semi-finals.
But James McClean’s goal secured victory for the Republic and sets up a two-legged play-off against one of the other group runners-up in November.
Meanwhile, Wales are lining up a friendly in November, with Mexico or the United States possible opponents.
Coleman added: “There’s a chance I can [stay] and a chance I won’t. I can’t give an answer right now.
“It’s not just a matter of signing a new contract. The FAW has been great with me. I asked them to leave me alone as my contract has been running down and all the questions were getting asked.
“I wanted to concentrate on getting results. There’ll be a conversation in due course.
“Right now, I’m thinking about the experience. I’ll go back to my family and take a bit of time. When the dust settles we’ll see where we go.”
Only Chris Coleman can answer the question over his own future. I think he will walk away now.
I do not think he will be short of offers. He will be lined up for a big job, if that is what he wants.
The Welsh players will be extremely, extremely down. Wales had to win their last four games – they won three and just came up a little short tonight.
Wales were not quite inventive and creative enough around the Irish penalty area. The Irish were outstanding defensively.