The Geordie has been linked with a return to Newcastle United in recent days.
"Andy will come good. He didn't have a pre-season last year," Sullivan told the Daily Mirror.
"He never really got his act together because of that and I think you'll see a much better player next season.
"It was a fair decision to leave Carroll out of the England squad based on what he did last season.
"He didn't do enough when he came back. There were a few games he was devastating in. Against Swansea he set up both of Kevin Nolan's goals. But he knows he didn't do enough to play for England.
"And I'm glad he is not playing for England. I don't want him arriving three weeks late because he has been in Rio all summer. But his time will come."
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