Unhappy fans, for whom Allardyce was not a popular choice when he was appointed in November, have been calling for his 18-month contract to be ended in the summer.
But the manager is taking a wider view of things, and asked whether he wanted to stay at the club he said: "Of course.
"I am having talks with him (Moshiri) but those talks are confidential.
"I came out of retirement because there was a long-term plan at Everton, a long-term aspect of going forward which I was excited to be a part of and understanding the history of Everton Football Club.
"It is a difficult period at the moment, yes, but we have to turn that around and start on Saturday.
"There are tough times and there are times when you have to draw in and close the doors and stay focused on what you want to try to achieve.
"It is my responsibility to get the team to play to the level they know they can play; players' responsibility on the field of play is to play their best and I think when they play their best we can give anyone a game."