Clear agreement on proceeding without the United States would be a boost for the principle of multilateral free trade pacts over the bilateral deal-making that Trump argues will give a better result for American workers.
Japan has been lobbying hard ahead of this week's meetings on the sidelines of the Asia-Pacific Economic Cooperation (APEC) summit to get full signoff on an agreement that is also a counterweight to China's growing regional dominance.
But Canada, whose economy is the second biggest among the TPP-11 after Japan, has joined other countries seeking changes or with misgivings about moving too fast.
"Canada will not be rushed into a deal that is not in the best interest of Canada and of Canadians," Prime Minister Justin Trudeau told a news conference in Vietnam's capital Hanoi.
The TPP aims to eliminate tariffs on industrial and farm products across a bloc whose trade totaled $356 billion last year. It also has provisions for protecting everything from labor rights to the environment to intellectual property - one of the main sticking points.
The original 12 countries had reached agreement on the TPP in 2016, but Trump withdrew in one of his first acts in office - throwing its very survival into doubt.
Trump and other APEC leaders, including President Xi Jinping of China and Russian President Vladimir Putin, will meet on Friday in the Vietnamese seaside resort of Danang.
Leaders of the TPP-11, all of which are also APEC members, will break away to consider proposals for proceeding with the trade deal that have been drawn up by their trade negotiators and ministers.
Japan had said it hoped a broad agreement could be reached among the 11 countries in Danang.
"This is not about Friday, it's about the next decades," Canada's trade minister, Francois-Philippe Champagne, told Reuters. "We are constructive, we are even creative at the table. At the same time we want to make sure we get the right deal."
The positions of both Canada and Mexico are also complicated by the fact that they are renegotiating the North American Free Trade Agreement (NAFTA) with the Trump administration.
Also in doubt is the appetite of New Zealand's new government to move ahead swiftly. Prime Minister Jacinda Ardern said it was too early to say a deal could be reached this week.
"We see a real responsibility to come in and make sure that the agreement is in the best interest of New Zealand," she said.
Both Malaysia and Vietnam would have been big beneficiaries of the original agreement as a result of the removal of U.S. tariffs. Malaysia, with an election approaching and growing ties to Beijing, has less incentive to move ahead fast.
Among options being discussed by TPP countries is whether to suspend some provisions of the original agreement to avoid having to renegotiate it and potentially to entice the United States back in the long term, officials said.
(Additional reporting by Ana Nicolaci da Costa in Wellington and Mi Nguyen in Hanoi; editing by Mark Heinrich)
By Matthew Tostevin and Mai Nguyen