Mr Kawczynski said he made his comments to Rishi Sunak after being invited to brunch at Chequers, the prime minister's country retreat, with his partner at the weekend.
It follows a meeting with Mr Campion last week in which Mr Kawczynski was trying to find solutions to the anti-social behaviour plaguing Shrewsbury town centre.
Mr Kawczynski said: "We have residents near the Ark frightened to leave their home or place of work, elderly residents who won't go out due to a cabal of people in Pride Hill."
The MP told Mr Sunak at the weekend that in his 18 years of being an MP he had never experience a worse situation in the town centre, and that he has now put Mr Campion "on notice".
He said: "The situation in Shrewsbury town centre and in Pride Hill is no longer acceptable. I have already spoken to the PCC about it. We need action in it. I have told the PM that I have now put the PCC on notice and that we have to have more visible policing in Shrewsbury town centre.
"We are giving him [the PCC] until next month to see a significantly improved level of police in Shrewsbury town centre. If things do not improve, I will be asking the speaker of the House of Commons to raise a debate in Parliament on the matter."
Mr Kawczynski added that other MPs have also raised concerns about town centres in their constituencies, but he said that in Shrewsbury it was his constituents' "number one issue".
"The PM listened carefully to what I said and I told him I would keep him updated with any decision about the PCC. He invited me to come back and discuss the issue with him if the situation does not improve," said Mr Kawczynski.
PCC John Campion said he shared the MP's concerns about Shrewsbury town centre.
He added: “As PCC, I am committed to working with partners to address the issues. This has been reflected in my office being part of the taskforce that has been set up by the council. I will also continue to ensure that West Mercia Police has the resources it needs to address these concerns.
"However, it is also important to make sure the wider system is set up to support those who are displaying the behaviours that are causing concerns and preventing these behaviours from being repeated in the future.
"I hope the action being taken, by all relevant authorities, will provide some reassurance to the communities of Shrewsbury that action is being taken.”