"You're asking for unconditional support and only my children get that," said Cosslett, who hailed England's performance on Saturday but said he would "keep a sense of proportion" about it, given the side's recent poor run.
"And, even then, not always. It's not something you give because you're under assessment the whole time. It's just what we do. There are 600 people in the RFU and we don't go around telling them they're all supported.
"They should know they're supported, assuming they're doing their job well enough. That's where we are. There's no point at which that stops or starts. He [Jones] has earned the right to make a strong case for the way ahead. We'll listen to his review on the tour and it'll be fascinating to see what he's got to say."
Asked if there was a cut off-point in which it would be too late to sack Jones, given the start of the World Cup is just 15 months away, Cosslett said: "You have to make a decision – life isn't like that. Things happen at the most inconvenient time – that is what life does.
"It always seems to happen like that. There is no time limit. And you make decisions on the basis of the evidence or information you have at any given moment. Hopefully we won't have to face that and we will have a run through to the World Cup.
"We have shown a lot of confidence in Eddie and we think he is still the man, but we will see where it goes to from here in terms of the next few months.
"There's no time limit. In a continuous review environment, you're continually trying to understand how you get better. Eddie will be making those points to Steve and Steve will take a view. We haven't had the review yet, so we don't know what will come out of it. Eddie's take on the tour will be fascinating. There's no moment when these decisions are taken."
Cosslett said the post-tour review would not be as extensive as the inquest into England's Six Nations campaign, but that it would be critical to planning preparations for the World Cup.
"It certainly won't be a review like we had after the Six Nations, that was a drains-up review of what took place," Cosslett added. "There were a lot of people involved in that. This is much more of a hand on the tiller, what's going on, what did we see down here, what can we improve on.
"Let's have a look at the plan from here and how does it stand because we have 15 months until the World Cup. Everyone knows that's the focus. Eddie has to come in and tell us his plan."
Part of the review will be the make-up of Jones's coaching team, with a successor to Paul Gustard, the defence coach who is leaving to take charge at Harlequins, expected to be announced next week.
Jones also confirmed that he was optimistic that Scott Wisemantel, the interim attack coach who had a positive impact on the tour, would become a permanent member of his coaching team.
"He did a good job on tour," Jones said. "He's mad as a cut snake, though, so if I can find a way of keeping a snake controlled, then maybe we might keep him.
"He's done really well and the players have enjoyed his coaching. It's a matter of whether he can fix his home situation, because he's got a family with three young kids. I'm cautiously optimistic that we'll be able to keep him on."
Jones said he had picked his World Cup squad on Friday, a regular exercise, and it was not "too different from what we had pencilled in before" despite the series defeat. "We have a pretty good idea who can go forward and who can't. A lot of those guys weren't on tour," Jones added.
