Leeds have since slipped to second off the bottom in the Premiership after defeats to Manchester United and Liverpool and then dumped out of the Carling Cup on Tuesday by the Old Trafford side.
The visit of league leaders Arsenal this week is another big test, although the last three performances have been very encouraging, despite the results.
McKenzie said: "There's no board meeting scheduled for some time now following the announcement of our financial results, but we are going to see how things go for a couple of matches.
"At any time when you are near the bottom of the table, any board is going to have a look at the situation.
"But we committed ourselves to him and then we recommitted ourselves to him and hopefully he will produce some good results for us.
"Peter initially came in at a difficult time, we gave him a chance and he did well for us. Hopefully he can pull us out of this situation too."
If results go against The Whites this week we could be bottom of the pile by Sunday.