Iain Duncan Smith said a decision by EU leaders to back Mr Juncker for the post would be "flicking two fingers" at voters across the Continent who had expressed concerns about Brussels.
The former Tory leader said the rise of Eurosceptic parties had sent a "seismic message" and accused heads of government of being "utterly complacent". Mr Cameron will call for a vote from fellow EU leaders unless his counterparts are prepared to consider an alternative candidate.
The move would mark a distinct break from the way the commission president is usually chosen, with the nomination agreed by consensus among leaders.
Polish Foreign Minister Radoslaw Sikorski said: "If the Tories were part of the European People's Party he could have made that argument ... when EPP chose its candidate, but EPP made its choice."