The force said it could not comment about which posts had been reported.
In his post on X, Mr Barton said two officers had knocked on his door at 21:30 BST on Thursday “when [his] kids were in bed”.
Mr Barton said in the post that he had given the officers his solicitor’s contact details.
He also said his solicitor had tried to contact a sergeant at the force but had been unsuccessful.
“Either it’s a shambles or an attempt to intimidate me and my family”, he added in his social media post.
A spokesperson for Cheshire Police said: “To date, we have not had any response from his legal representatives and we await further contact.”