OTTAWA—Canada’s border agents started a partial strike on Friday that threatens the flow of U.S.-Canada trade and could upend a plan to allow American tourists starting Monday to visit their northern neighbor for the first time in 16 months.

Overnight talks between the union representing about 9,000 employees at the Canada Border Services Agency and the Canadian government failed to produce a new agreement before a 6 a.m. Eastern time Friday deadline.

The Public Service Alliance of Canada, which represents customs and immigration officers, said front-line agents would scale back the amount of work conducted at land crossings and airports and warned the work slowdown could cause long and unavoidable delays for shippers and travelers at Canada’s borders.

A union spokesman said the bargaining team has been in mediation with government negotiators “all night and through to this morning, and we’re giving them a bit more time to negotiate at the table.” The spokesman said work-to-rule actions are now under way at border crossings and airports across the country.

“The government is still at the table and will not walk away,” a spokeswoman for Canada’s Treasury Board, in charge of public-service staffing, said Friday. A mediator was brought in midweek to help resolve differences, which the union says focus on working conditions and compensation.

The border agency said this week that it was ready to respond to possible labor disruption to maintain border security and keep the border open to legitimate travelers and goods.

The Canada Border Services Agency’s unionized employees have been without a contract for about three years. Two-thirds of those workers are classified as essential and unable to go on strike. Instead, the union said these workers would conduct a work-to-rule campaign. For instance, border agents won’t work past scheduled hours or during their allotted breaks, and will slow down the processing of trucks and travelers and refrain from activity that isn’t required under the job description.

Video posted to social media show a lengthy lineup of trucks stuck on Interstate 75 in Detroit, en route to the Ambassador Bridge which connects Michigan to Canada. The Department of Transportation said the bridge is the single busiest international land-border crossing in North America, accounting for 27% of the roughly $400 billion in annual trade between the U.S. and Canada.

The start of the partial strike “will hamper manufacturers’ ability to get the essential components and goods to sustain global supply chains and threatens thousands of Canadian businesses,” said Dennis Darby, head of the Canadian Manufacturers and Exporters, a lobby group.

The partial strike comes days before Canada lifts some of the border-entry restrictions first imposed in March 2020 to help limit the spread of Covid-19 in the country. On Monday, fully vaccinated U.S. citizens living in the U.S. will be permitted to enter for tourism and recreation purposes, subject to testing.

Business groups, who pressed for the Canadian government to relax border measures, worry the possible labor disruption could dissuade American tourists from coming to the country.

Canadian businesses in the travel, tourism and hospitality sectors, whose sales plummeted because of the Covid-19 pandemic amid border restrictions in place since last year, were relying on a partial reopening at the U.S.-Canada border to help salvage what is left of the summer season.

Canada plans to open its borders to fully vaccinated travelers from countries other than the U.S. in early September.

Write to Paul Vieira at paul.vieira@wsj.com