Nicola Poultney, who, says the paper, lives with Albert Bore, secretary of the ruling Labour group on the council, has been promoted to senior public relations officer.
John Alden, deputy leader of the council's Conservative group told the paper: 'I will ask for inquiries to be made about why this happened and whether the council's standing orders have been scrupulously applied.'
A council spokeswoman said: 'We have a clear and open appointments procedure which is always followed.
'The procedure was followed, and it was an open process in competition with external candidates in competition with external candidates after external advertisement. The best candidate was appointed.'