The loss against Ireland was particularly hard to take for Coleman, as it marked the first defeat Wales have suffered in a competitive fixture since they lost 3-0 to Serbia in September 2013 in the qualifiers for the 2014 World Cup.
Roberts, who has been Coleman's assistant since he was appointed Wales manager in January 2012, added: "He couldn't have taken on the job at a more hard time and what we've collectively achieved since then has been incredible".
Speaking to BBC Radio 5 Live after the game, Coleman's former worldwide teammate John Hartson, said he may step away from the job.
Wales players decided they needed to approach Coleman, and tell him of their wish for him to stay.
A group of the more experienced members of the Wales squad held a meeting with Coleman immediately after the defeat to Ireland to express their wish that he stays in the post he has occupied since January 2012.
"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".
The 28-year-old has been a regular feature in Coleman's side since he took over the reins from his predecessor Gary Speed.
"There's a chance I can stay and a chance I won't, I can't give an answer right now", said Coleman. It's not just a matter of signing a new contract, the FAW have been great with me. There will be a conversation in due course and there's a friendly next month. "Why you'd want him to leave I'm not too sure".
"I know I'm going to be asked about my situation but I won't think about that right now".
Coleman said Wales "just came up a bit short" and added: "It's a shame for us, a hard one, but we've got no complaints".
"The players coming through will strengthen what we have, that excites him and is obviously a major factor".
Ireland are through to the qualifying playoff thanks to a goal in Cardiff from James McClean, which also eliminated Wales.