The Tory Minister rejected calls from his LibDem Coalition partners they should have the final say.
The LibDems fear Mr Gove plans to give the top job to a leading Conservative donor.
A party source for the junior coalition partner said earlier this week that there was no way "Tory donor ideologue" Theodore Agnew should be given the job.
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But Mr Gove, the Westminster Education Secretary, said that "no politician will have a responsibility for finding a replacement".
He said there would be a "rigorous independent process", but that ultimately he would decide whether or not to accept any independent recommendation.
The row centres on Mr Gove's decision not to award a second term to Labour peer Baroness Morgan of Huyton.
Labour have called for her to be reappointed as chairwoman of the education watchdog Ofsted.
LibDem leader Nick Clegg wants his party's education minister, David Laws, to appoint her successor.
Meanwhile, Labour MP Michael Dugher has written to the Cabinet Secretary Sir Jeremy Heywood over Baroness Morgan's removal.
Mr Dugher wants the UK's top civil servant to investigate whether there is now a "pattern" of sacking non-Tories from top posts. He wrote: "Many rightly see this as an emerging pattern of behaviour in Whitehall with non-Conservative supporters being replaced by prominent Conservatives, including major donors, to powerful public-sector roles to support the Government's political agenda."
Mr Gove was forced to defend his department saying it was not a "gallery of nodding dogs" after a warning he should not surround himself with "yes men". He said that officials in his department said, "yes, no, maybe" to ministers.
His comments came after Sir David Bell, vice-chancellor of Reading University, warned the minister in an online blog not to surround himself with "yes men".