MP Clive Betts has suggested the change as a "pragmatic" way of making different factions in the party "work together", denying it is an attempt to "hobble" Jeremy Corbyn's leadership.
Appointments have been the leader's responsibility since 2011 when the elections system was scrapped.
Labour's ruling National Executive Committee must approve any changes.
A spokesman for Jeremy Corbyn has suggested any debate about changing the rules should consider whether MPs, conference or the party membership decides.
A secret ballot of MPs will be held on Tuesday after the issue was discussed earlier at a meeting of the parliamentary Labour party. If MPs vote for the change, the NEC may refer it to party conference to decide later this month.
Labour is currently in the middle of a leadership contest, with Owen Smith challenging Mr Corbyn's position. The outcome will be announced on 24 September.
Ed Miliband ended elections for Labour's front bench in 2011, a method of selecting the leader's top team in opposition which had been in place for several decades.
Mr Corbyn, who opposed Mr Miliband's decision at the time, chose his own top team after his election as Labour leader last year but more than half of them resigned in protest at his leadership in the wake of June's EU referendum.
In a subsequent confidence vote, only 40 MPs supported Mr Corbyn's continued leadership while 172 said they could not back him.
Mr Corbyn's spokesman said he supported "democratisation and reform of the party rules and structures".
He added: "How the shadow cabinet is made up is one part of the debate, including whether part of it should be elected by MPs, by members or by conference."
BBC political correspondent Glenn Campbell said Mr Smith was thought to be supportive of the idea but had yet to publicly come down on one side or another.
Labour MPs are to vote on Tuesday on a proposal to allow them to elect members of the shadow cabinet.