The bill has been the first cause of serious dissent in the ranks of President Emmanuel Macron's party.
Interior Minister Gérard Collomb, who attended almost all the discussions, hailed the committee's attempts to "satisfy the need for efficiency and the desire for humanity" on a "difficult text that brings into play a certain number of values".
But two REM MPs, Matthieu Orphelin and Stella Dupont, said they would be tabling "indispensable" amendements during the parliamentary debate.
Right to work
Much of the discussion so far has centred on a proposal to allow asylum seekers to work after six months, instead of nine as the law states at present.
Collomb said the government was in favour in principle but "asylum seekers should not be able to profit from an easier procedure than people who have come to France to work legally".
Although the measure is already in effect in Germany, right-wing MPs are vehemently opposed to it, arguing that it would lead to failed asylum seekers being granted the right to stay in the country.
Former Socialist prime minister Manuel Valls, who supports the Macron government, warned that it could lead to problems for employers who have taken on workers whose applications eventually fail.
Another sensitive question is the "crime of solidarity", which has seen court action against campaigners who help migrants enter France out of personal conviction.
The REM vice-chair of the committee, Lois Naïma Moutchou, drew up an amendment to end legal action against people who have helped migrants "without seeking financial gain and without direct or indirect material reward" but agreed to redraft it for the submission to the plenary session.
Collomb expressed the wish to distinguish between "occasional" helpers and "an extremely dangerous category of totally irresponsible people ... who are calling for the abolition of borders".