Mr Juppe called on both Mr Cope, the party's secretary general since 2010 and Mr Fillon, prime minister under Mr Sarkozy, to "accept the decision of the electoral commission when it is delivered".
Mr Fillon, who was prime minister under President Nicolas Sarkozy until his electoral defeat in May, has likened Mr Cope's allies to "a mafia" while his rival has described him as a "sore loser".
As left-wing bloggers mocked Mr Cope's victory, dubbing the party under his leadership the "FNUMP", Mr Cope said he had telephoned Mr Fillon to ask for his support.