Layoffs have begun to sweep through Target Corp., even as the company’s CEO assured Gov. Mark Dayton Monday that the retailer will maintain a robust headquarters in Minneapolis after the cuts.

Some vice presidents were terminated last week, employees told the Star Tribune, in the first of what the company has said will be “several thousand” job cuts.

One employee who declined to be identified said that probably a couple of dozen upper managers were let go Monday and gone by lunch. The employee said broader cuts were expected Tuesday.

“The mood is pretty somber, about what you would expect,” the employee said.

Target CEO Brian Cornell met Dayton, whose father and uncles created Target Corp., for about 45 minutes Monday at Target headquarters. Also present were Lt. Gov. Tina Smith and top Target executives Jodee Kozlak, chief of human resources, and Tim Baer, chief legal officer.

“I asked for an assurance that they were committed to Minnesota, to Minneapolis, to keeping their headquarters here, their general operational headquarters,” Dayton told reporters after the meeting. “I got strong reassurance that was their intent and the case, and they never considered anything else.”

Cornell last week revealed plans to cut “several thousand” employees and said his goal was to create a “much more agile, effective organization.”

The company has not said exactly how many people will be cut, nor in which departments, but has said the process could take two years. Target declined to elaborate on Monday.

The company employs about 13,000 people at its Twin Cities headquarters, including about 10,000 in downtown Minneapolis.

Dayton said he was told that savings from the downsizing will be reinvested in the company, “much of it here in Minnesota.” The governor also said he came away from the meeting confident in Cornell’s leadership and his team.

“They inherited this situation,” Dayton said. “He’s been there less than a year, and he walked into this.”

The previous CEO, Gregg Steinhafel, departed last May amid operational difficulties, a mishandled expansion in Canada and a data breach that put customer information at risk.

Cornell decided in January to end the Canada expansion, which produced more than $2 billion in losses.

As a result, 17,600 workers in Canadian stores are losing their jobs along with 550 who supported them from corporate offices in the Twin Cities.

Cornell told the governor on Monday that the latest job cuts will “help ensure that Target continues to grow and thrive in Minnesota” and that the company aims to “treat our team with the respect they deserve,” according to a short statement from Target.

Dayton said Cornell suggested a follow-up conference call at the end of the week.

“They’ll update us further on developments and continue the dialogue,” Dayton said.