Gavin Callaghan, councillor for Pitsea North West, claims he has been told to quit the party by Labour activists in Basildon.
Smith said Corbyn had led the party to the brink of disaster.
Last year, shortly after Corbyn won an overwhelming mandate from Labour party members, Foster stood at the back of a Labour Party conference fringe meeting in Brighton shouting: "Say the word "Israel, ' say the word 'Israel, '" after Corbyn addressed LFI supporters".
Nominations for the election close on July 20, and there is a two day window between July 18 to 20 for people to pay £25 to register to vote in the contest, with ballot papers being mailed from August 22.
The NEC on Tuesday agreed that Mr Corbyn will automatically appear on the leadership ballot but introduced restrictions on who will be able to vote in the contest, which his rivals hope will benefit them.
Yesterday, after six hours of negation, Labour's National Executive Committee ruled Mr Corbyn should be included on the ballot despite opponents arguing the leader needed 51 MPs to support him.
"Jeremy is all about bridge building and he will work to unite the Labour party under his leadership so, of course, Jeremy won't introduce mandatory reselection for sitting MPs and he has absolutely no intention of deposing sitting MPs", said the spokesperson.
Numerous senior Labour officials accused their leader Corbyn of a lackluster campaign to keep the country in and triggered an election process to challenge him.
"All I am simply saying is, the advice that was taken was certainly not given the expert consideration that it would receive from a High Court judge".
The move could help those who wish to remove Corbyn.
The MP for Pontypridd was due to launch his campaign this morning in his constituency to make his case over current leader Jeremy Corbyn and fellow former Labour frontbencher Angela Eagle.
The NEC decided the leader did not have to collect support from MPs and MEPs as other candidates do, because he is already leader.
He explained: "It's fair to say we were probably a little nervous about going onto the streets of Chelmsford, a city that has elected a Conservative MP since 1950".