François Legault, the premier of Quebec, has fretted publicly about the decline in SNC-Lavalin’s share price, which has been partly attributed to the legal uncertainty surrounding the company. He has vowed to block any takeover prompted by that low share price to keep SNC-Lavalin headquartered in Montreal.

In 2014, a former senior executive at SNC-Lavalin pleaded guilty in Switzerland to bribing Mr. Qaddafi’s son, Saadi. Last month, Pierre Duhaime, the former chief executive of SNC-Lavalin, pleaded guilty to “willful blindness” when the company paid 22.5 million Canadian dollars, or about $16.9 million, to hospital managers in Montreal for information it used to win the contract for a new building.

SNC-Lavalin has been running a public relations campaign to gain public support for a settlement to the Libya case, arguing that it had reformed its internal practices and purged its ranks of corrupt executives.

The possibility of a remediation agreement was made possible by a measure introduced by the last budget legislation from Mr. Trudeau’s government. Opposition members criticized the government for burying it within the budget rather than presenting it as separate legislation.

Cameron Ahmad, a spokesman for the prime minister, said in an interview on Monday that the measure was discussed by parliamentary committees.

Mario Dion, the ethics commissioner, said in his letter that he would investigate Mr. Trudeau’s actions under a section of the Conflict of Interest Act that prohibits federal politicians from using their influence to “improperly further another person’s private interests.”

Speaking in Vancouver, British Columbia, on Monday, Mr. Trudeau said he welcomed the inquiry.

“This is an issue that has been much talked about over the last few days, and I think it’s important Canadians continue to have confidence in our system,” he told reporters. He added that in conversations with Ms. Wilson-Raybould last fall, “I told her directly that any decisions on matters involving the director of public prosecutions were hers alone.”