Lingard, who initially joined the Blues on a one-month loan, made a dream debut in front of his new home crowd on Saturday, scoring all four goals in their 4-1 victory over Sheffield Wednesday.
The 20-year-old attacking midfielder completed his hat-trick on the half-hour mark before netting a fourth six minutes into the second period as he left the Owls defence in disarray.
Following the exceptional start, Clark is eager to extend the deal so that Lingard can continue to play a pivotal role as the Midlands-based side look to climb from 18th in the Championship standings.
"What we have got to try and do is continue to win games and for Jesse to enjoy his football to make the stay longer," Clark told the Birmingham Mail.
"That's what we aim to do because, you know, we certainly don't want him here just for the short term. We want him here for as long as possible.
"The month will be up quite quickly. Manchester United need to see that he is developing as a player and is enjoying the environment where he is playing."
Clark did also confirm that he has been in contact with United manager David Moyes regarding Lingard's opening display.
"I sent a text to David Moyes after the game Saturday," he added. "I see by the television footage that he was at the Hawthorns.
"He just replied saying he was delighted for the lad."