Having turned 28 in July, and with his fortunes tied to a country that last qualified for the World Cup in 1958, Bale has missed out on his best chance of playing at football's biggest event.
The squad have given their unanimous support to Coleman and it is understood the Football Association of Wales is also eager for him to extend his near six-year stay at the helm.
Gareth Bale, together with other members of the Wales squad will make a concerted effect to persuade Chris Coleman to remain as national team coach in spite of the team's failure to qualify for next year's World Cup finals in Russian Federation.
The 47-year-old said in the immediate aftermath of the Ireland defeat he would hold talks with the FAW over his future.
"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".
"He couldn't have taken on the job at a more hard time and what we've collectively achieved since then has been wonderful".
There were high hopes that Wales would go on to qualify for the 2018 World Cup in Russian Federation but that would not be the case. "What he has done for the team and Welsh football has been fantastic".
Coleman, who led Wales to a stunning semi-final finish at Euro 2016, is out of contract at the end of the season and he was evasive when asked if he might stay on.
But his stance appears to have softened in the last 12 months and Coleman could be open to the idea of changing his mind and leading Wales into the Euro 2020 campaign.
The Real Madrid man said: "Of course we want him to stay. Why you'd want him to leave I'm not too sure".
"The whole nation will be mourning and disappointed because, again, that elusive World Cup has passed us by. When the dust settles we'll see where we go".
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