It is claimed Councillor Barrow is currently under investigation by the local authority over a complaint that he failed to comply with the council's code of conduct.
Deputy leader of the council and fellow Conservative councillor Steve Charmley said: "I am aware of an investigation. It is an ongoing investigation as I understand it.
"I am sure that Keith will be vindicated or cleared. I would not want to comment on it at the moment as it is an ongoing investigation."
The council's code of conduct, which is available to view on its website, says members must follow a number of principles – "selflessness, integrity, objectivity, accountability, openness, honesty and leadership".
The code says members should act "solely in the public interest" and be as "open as possible" about decisions and actions made.
Shropshire Council said it could not confirm or deny that an allegation had been made.
Claire Porter, corporate head of legal and democratic services, said: "Following receipt of any allegation regarding member conduct, and until the conclusion of any investigation, Shropshire Council will not confirm or deny that an allegation has been received due to the risk that the process may be compromised.
"All allegations that are received are dealt with in accordance with the council's arrangements."
Council spokesman Adam Spreckley said that Mr Barrow is currently on holiday and cannot be contacted.