The prime minister's spokesman, Andrew MacDougall, says the two sides are down to a few outstanding issues.
"We've been at the negotiating table for quite awhile now, trying to get the deal that's in Canada's interests," MacDougall said Friday.
"I tell people it's not like we're looking to swap sixth-round draft picks here. We're actually trying to sign the most comprehensive trade agreement that Canada has ever signed, that involves multiple levels of governments, and we're not there yet.
"We're down to a few outstanding issues. I won't speculate or comment on what they are, but negotiations are ongoing, and I don't expect that we'll be in a position to sign a deal next week."
Canada is under pressure to conclude a deal before the European Union turns its attention to free-trade negotiations with the United States this summer.
NDP Leader Thomas Mulcair says he's concerned Harper will sign a bad deal out of desperation to wrap up negotiations before the Europeans shift their focus to the American talks.
"The talks that have been going on for some time now seem to be leading to a bit of an impasse," he said.
"But we're quite concerned that in the current context, the desperation of the prime minister leads him to sign a bad deal. Desperation is a singularly bad adviser."
Among the issues believed to be on the negotiating table are financial services, Canadian beef exports, country-of-origin rules for vehicles, procurement limits for provinces and municipalities and drug patent protection.
Laura Dawson, an Ottawa-based trade consultant, said the European talks as they currently stand appear similar to how Canada's free-trade negotiations with the Americans looked near their conclusion in the late 1980s.
"If it's true that we're just down to dairy market access and beef market access, I think that is a really positive sign that this is a high-quality agreement and they've gotten through a lot of other, complicated negotiating subjects and that they're both holding fast on issues that are of national, economic importance," she said.
"This is much more difficult than negotiations with Panama or Peru, because we're dealing with advanced, industrial economies who have a very challenging set of demands. Our template for how we negotiate trade agreements comes from the 1994 NAFTA, so we've really had to step up our game, and the EU is no slouch."
Harper departs for Europe on Tuesday and will spend the week in England, France and Ireland before joining leaders from the world's eight richest economies for a two-day summit at a lakeside Lough Erne resort near Enniskillen, Northern Ireland on June 16.
British Prime Minister David Cameron wants the G8 to focus on fighting tax evasion by multinational corporations and wealthy individuals.
Some have suggested Canada is not fully in favour of Cameron's call for more transparency about tax information and the ownership of companies. But a Canadian government official who spoke to journalists at a pre-G8 briefing on the condition his name not be published insisted Canada and the United Kingdom are not that far apart on the tax issue.
"I think at this point in time it's safe to say that we're very well aligned with the U.K. in broad terms on actions to combat tax avoidance, tax evasion and promote corporate transparency," the official said.
"We won't comment on summit negotiations around language, but I think in broad terms we're very well aligned in supporting the U.K. in their leadership on this issue and taking actions in Canada."
Beyond talks about the economy and financial issues, the G8 is also expected to focus on international security, particularly the Syrian conflict, Iran and North Korea's nuclear programs and anti-terrorism measures in Africa's Sahel region.