After a poor start to the season, the North London side had turned around their form to rise from the lower part of the table into the top four but Sunday's 3-2 defeat by promoted Swansea City and a 2-1 loss at Fulham have stalled the progress.
Arsenal are in fifth place, 12 points behind the two Manchester clubs who are level on points at the top before City's trip to bottom club Wigan Athletic later on Monday.
"That's now twice - at Swansea and at Fulham - where we have given the game away," Wenger was quoted as saying by the club's website.
"At our level that is frustrating and not good enough. It is down to a lack of composure and the right decision-making.
"It is alarming because at Fulham we were 1-0 up and we lost the game. [Against Swansea] we came back to 2-2 and straight away we gave the third goal away. It is really a concern."
Arsenal have lost seven league matches this season and are determined not to add to the tally when United visit Emirates Stadium next Sunday.
"It is a massive opportunity because you feel we have given six points away in the last two games in a way that is unbelievable," Wenger said.
"It is time for us to wake up and be focused - we have a good opportunity to do that against Manchester United."
Arsenal, whose below-par season has coincided with arch rivals Tottenham Hotspur challenging the Manchester clubs for the title, have been boosted by the short-term loan signing of their all-time leading goal-scorer Thierry Henry.
The Frenchman, who plays for Major League Soccer outfit New York Red Bulls, enjoyed a dream return when he came off the bench to score the winner in their FA Cup win over Leeds United last week but even that has soured a little.
Monday's British newspapers reported Henry had been involved in a heated verbal exchange with an Arsenal supporter after the Swansea defeat when he gestured at the fan and urged him to be more positive and get behind his team.
