A long-running feud between the two cousins culminated in the ouster of Arthur T. Demoulas as CEO in June. His firing prompted worker protests and customer boycotts that have brought the private company to a virtual standstill.
The company has lost millions of dollars over the last four weeks as warehouse workers have refused to make deliveries and customers have shopped elsewhere in a show of support for the employees, leaving the stores with only a trickle of business.
Arthur T. Demoulas has offered to buy the 50.5 percent of the company owned by his cousin and other relatives on his side of the family.
Heather Nichols, a spokeswoman for Patrick, said the two sides made "real progress" on the terms of the sale and operating control of the company during the negotiating session. She said the governors are encouraged that a resolution may be "within reach."
Hassan's press secretary, William Hinkle, said the governor is hopeful that the leaders at Market Basket can resolve the standoff.
"Market Basket is a significant business in New Hampshire, and its role as an employer for thousands of dedicated employees and as a provider of affordable groceries for its customers impacts our economy and communities in many ways," Hinkle said in a statement.
Neither Demoulas cousin commented.
The company's board of directors was scheduled to meet Monday.
Last week, the company warned warehouse workers to return to their jobs by Monday or face termination.
Tom Trainor, a district supervisor who worked for Market Basket for 41 years before being fired last month for helping to organize the employee revolt, said that the approximately 300 warehouse workers and 65 drivers who have refused to make deliveries for the last month remained off the job Monday, despite the warning.
"They're still solid. They are not coming back [to work] until ATD (Arthur T. Demoulas) comes back," Trainor said.
"It's definitely encouraging that the two governors have realized that they're stakeholders in this. Having pressure from their offices is certainly welcomed by our wide," he said.