Coleman, due to see his contract expire after Tuesday's friendly with Panama, reiterated this week that he intends to make a decision over his future by the end of November.
The former Coventry City and Fulham boss guided Wales to their first tournament finals appearance in 58 years last summer, defying all expectations by reaching the last four of the European Championships.
Gunter believes that stability is now the key to pushing on, having disappointingly missed out on a World Cup 2018 place, with each and every member of the squad desperate for Coleman to stay on.
"Whenever a player gets asked the question in an interview, the line is always the same: you want the manager to stay," he told reporters. "But, with us, I can't understand – and I wouldn't know – one fan, one player, one person in Welsh football, who wouldn't want him to stay. He is the greatest manager in Wales's history, the things he's done for this country is incredible. And the last thing this group of players need is for him to leave.
"Truthfully, hand on heart, I can promise you it's the same from every member of the squad, we want him to stay, it would make absolutely no sense to let Chris Coleman leave Wales. We need him to stay, that's what every single player wants – and it's not your stock line, that's God's honest truth. As a manager he's very good, talented, with very good knowledge and tactically we've been very good, which is important.
"But the thing that separates him from a lot of managers is the way he is a person. The atmosphere he's created in the squad and in the team hotel is the best it's ever been. Fans and media only see the end result on a match day, but for us in the hotel seven-ten days at a time, you can't just say we need a good atmosphere and a good feeling in the squad, it has to be made but also be natural. He is the main reason it's like that.
"Yes, we've got good characters and we all get on, but it all comes from him and his staff. That is the difference. I know players always say about this about their manager, but bring any player to speak about him, if it's ever announced he's staying, it would be the best bit of news we could have as a squad."
Gunter is in line to earn his 85th cap for Wales against Panama, equalling the late Gary Speed's record for outfield appearances.