The Blues boss has previously hinted he will be in charge at Goodison Park next season, but during his latest press conference, he conceded that there could be truth to the rumours that the Premier League outfit's board are considering a change of manager.
Allardyce, who guided Everton to eighth place in the league this season, is expected to be given clarity on his future when he meets with the club's hierarchy over the next week.
"Confident? I can't quite say after all the rumours I'm reading in the paper because generally there's something and there's no smoke without fire, but I'll wait and see when we meet up next week, or this week actually," he told reporters.
Everton ended their 2017-18 campaign with a disappointing 3-1 loss at West Ham United.