It comes after former Pontesbury parish councillor Paul Bradbury resigned from his seat after expletive-laden racist posts were made on his Facebook page.
Police have now confirmed a man in his 60s from Minsterley, who was arrested yesterday, has been released.
Chief Inspector Mark Reilly said: “We want to be absolutely clear - we have a zero tolerance policy for discrimination in all forms, including online abuse. We will investigate reports and act.”
A deluge of vile abuse was aimed at England stars Marcus Rashford, Jadon Sancho and Bukayo Saka, after the trio missed their spot-kicks in the penalty shoot-out against Italy on Sunday night.
One post on Mr Bradbury's page questioned the Black Lives Matter movement and another said: “Whites only in future England teams”.
The statements sparked outrage online, with one branding the words used “absolutely disgusting”, and another saying they “couldn’t believe” what had been written.
The BBC has reported that Mr Bradbury says his account was hacked. Mr Bradbury has not responded to the Shropshire Star's repeated requests for a comment on the situation.
Pontesbury Parish Council’s leadership held a meeting with police on Monday afternoon. The authority confirmed Mr Bradbury's resignation, and distanced itself from his "strongly offensive" comments.
A spokeswoman said: “Pontesbury Parish Council is aware of comments made on Facebook by a former member of the parish council. They are made by an individual not representing this council. We found these comments strongly offensive, and do not tolerate racism of any kind.
“The council have reported this serious matter to the police and are currently assisting them with their enquiries. The council distances itself from these comments. The chairman has accepted his resignation from Pontesbury Parish Council.”
Police urged anyone who has been a victim of hate crime to get in touch via its online reporting function or by heading to report-it.org.uk/.