Mr Coetzee is often referred to as the Liberal Democrats' equivalent of the Tories' polling guru Lynton Crosby.
But unlike Mr Crosby, Mr Coetzee is paid his £110,000 salary, the highest of Mr Clegg's special advisers or spads, from public funds.
His official title is director of strategy and he spends much of his time advising the deputy prime minister on how his party can boost poll ratings. The party insists Mr Coetzee fully complies with the special advisers' code of conduct. It explained his role entailed ensuring all Coalition messages strictly reflected Lib Dems' priorities. "He is a special adviser so he is paid out of public funds. The salaries are all transparent and published on the Internet," said a spokesman.
But Ms Gilmore pointed to the code, which stresses spads "should not use official resources for party political activity".
"If Mr Coetzee is working on issues concerning the Liberal Democrats' election strategy rather than the objectives of the Cabinet Office, where he is employed on the public payroll, this would appear a breach of the code," wrote the Labour backbencher. She added: "It would rightly be extremely concerning if the Liberal Democrats were exploiting public resources and the important role played by special advisers to further party political interests rather than Government objectives."