Ian Murray said the move could act as an “olive branch” to win over MPs who have been critical of the veteran left-winger, who looks almost certain to be re-elected as Labour leader on Saturday.
Mr Murray was one of a group of MPs, disgruntled with Mr Corbyn’s leadership, who quit the shadow cabinet en masse.
The former shadow Scottish secretary said he “certainly would” return to that post if shadow cabinet elections were restored, but added that not returning to the election system “makes it much more difficult”.
Labour’s ruling National Executive Committee has put off a decision on MPs’ demand to vote on who should be members of the frontbench team.
But after a divisive leadership contest, Mr Corbyn has pledged to “wipe the slate clean” and work with critics within the party if re-elected.
Mr Murray, MP for Edinburgh South, told BBC Radio Scotland’s Good Morning Scotland: “I think Jeremy has to make sure he does wipe the slate clean if that is his intention.
“He has to stop using de-selection lists in his office. And the PLP (Parliamentary Labour Party) have actually already said we want some unity in the party, and we have put out the olive branch of shadow cabinet elections, and I think Jeremy should accept those shadow cabinet elections as a an olive branch and a way of getting unity within the party.”
Mr Murray said elections to the shadow cabinet are “something Jeremy Corbyn used to support”, adding that the leader had been “one of the biggest opponents of removing shadow cabinet elections”.
He added: “I think if this is a way of uniting the party it’s a good start, it’s an olive branch, and I would suggest Jeremy grabs it with both hands.
“It’s about uniting the party, this is what we all want to do come Saturday, regardless of the result. We have to bring together all elements of the party – that includes the trade unions, the socialist societies, the Members of Parliament, the members of the party.”
He said the distinction between being appointed to the shadow cabinet and being voted on to it is “you would be elected by your peers”.
Mr Murray said: “We want to try to unite the party. If he doesn’t take on shadow cabinet elections he has no intention of wiping the slate clean.”
He also insisted there would be “absolutely no split in the party” if Mr Corbyn was re-elected, adding: “The party is far too important not just for the people who have been in it for an awful long time but for the values it represents and indeed for the way it wants to represent and govern the country.
“There will be no split in the Labour Party and I would suggest to anyone who wants a split that is the wrong way to go.”