The England U21 international linked up with the Blues on a one-month deal ahead of their home date with Sheffield Wednesday.
Lingard's debut outing for the Championship club could not have gone much better, as he netted all four of City's goals in a 4-1 win - including a first-half hat-trick.
The promising frontman, who impressed for United over pre-season but is yet to make a competitive appearance for the club, believes his ongoing development would now be better served if he was allowed to stay on at St Andrew's.
He said: "It's a dream come true. To get four goals and three points on my debut is massive.
"I had a lot of pressure on my shoulders but as soon as the first goal went in I got more confident.
"You can't beat first-team football, so hopefully they will let me stay for the season."
Birmingham would certainly like to keep Lingard on their books for the foreseeable future, with assistant manager Terry McDermott saying: "Jesse was absolutely unbelievable.
"It disappointed me that he had to come off because I think he could have scored five.
"Hopefully we will get to keep him."