Carver was appointed until the end of the season in January after former boss Alan Pardew left for Crystal Palace.
But with Newcastle stuttering towards the end of another mid-table campaign, Elliott has urged owner Mike Ashley to bring a fresh face into the dugout.
"I think it's easier for Ashley to keep John in the role but I don't know if that's the right thing for the club," Elliott told Perform.
"I feel we need a clean break and it's a very difficult role. Look at the managers we've had over the years - how successful they have been at other clubs, then they come to Newcastle.
"You'd like to think it'd be a job that interests a lot of people because of the possibilities.
"It might need a change of structure with how the owner wants to do it, that may prove a stumbling block, but it's going to be a big summer and I think that will dictate a lot of whether players stay or go and who comes in."
Newcastle released their accounts for 2013-14 this week, which showed a record annual profit of £18.7 million.
Ashley has been criticised for a perceived lack of investment in players, with Elliott suggesting the upcoming close-season period could be pivotal for the club.
"You can say this is a strange season but I think this is how the last few have panned out," he added.
"Obviously questions are being asked about upstairs and the investment but it's down to the players and the staff to basically get on with it.
"It's been very up and down. There's always something going on at the club. When it's quiet you're just waiting for the next thing to happen.
"We've got an opportunity to hopefully have a good end to the season, players are playing for contracts and their futures really.
"It's an important time for the club."
