This is expected to be decided at a meeting on Monday, which promised to be explosive.
Analysts say that offering a free vote to appease the rebels could leave Corbyn looking like he cannot control his own party.
Ahead of the meeting, the Labour Party said it had received tens of thousands of responses to a call by Corbyn last week for members to say which way they think the party should vote.
An initial analysis of 1,900 responses showed 75 percent opposed to bombing, 13 percent in favour and 11 percent undecided, party officials said.
This survey has sparked fury among some, who accuse Corbyn of underhanded tactics by rallying support for his own position instead of waiting for the party to make a collective decision.
"How does Jeremy Corbyn and his small group of tiny Trots in the bunker think they’ve got the unique view on it all?" senior Labour MP John Spellar fumed on BBC radio on Friday.
Another Labour lawmaker, Paul Flynn, told the BBC there were "terrible divisions" while a third, Fiona Mactaggart, called Corbyn’s position "unsustainable".
But one of his closest allies, Diane Abbott, on Monday urged Corbyn to "show leadership" and impose his line on the party.
She said that "the party as a whole" was opposed to the bombing.
Key to the row is the fact that Corbyn -- a political outsider and serial rebel before becoming leader -- was elected in September thanks to grassroots left-wing support, but is not widely backed by generally more centrist Labour MPs.
If Corbyn does impose a three-line whip and try to make Labour MPs vote against air strikes, Honeyman predicted that some of his frontbench team could quit.
"If you’re a member of the shadow cabinet and you defy a three-line whip, there’s an expectation that your resignation will be on the desk in the morning," she said.
While that would be deeply embarrassing for Corbyn, it is not clear it would herald the end of his leadership.
The process for removing a Labour leader is complex and Corbyn has a huge mandate from activists, making any challenge to him problematic. "I’m not going anywhere," the leader said on Sunday.