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The company could either suspend production or further reduce the production level, the source said.
"If they make the decision (to suspend) it's just a temporary pause of the line," the source said, adding that the production will eventually resume again.
This comes after a meeting last week between the Federal Aviation Administration (FAA) and Boeing in which the company said it came away understanding the 737 Max jet would not be cleared to fly this calendar year.
The planes were grounded worldwide in March after two fatal crashes — a Lion Air jet into the Java Sea in October 2018 and an Ethiopian Airlines plane near Addis Ababa in March — killed 346 people.
Also last week, Congress heard from a Boeing whistleblower who worked on the 737 Max and had urged managers to shut down the production line because of mistakes and cut corners. He said his recommendations fell on deaf ears.
An FAA analysis dated December 2018 — weeks after the first crash — predicted there would be more than 15 additional fatal crashes of the Max over its lifetime. That document was made public at a House Transportation Committee hearing on Wednesday.
The federal agency did not ground the planes until after a second crash.
Defending the agency's decision making, FAA Administrator Stephen Dickson said "any indication that any level of accidents are acceptable is not reflective of the 45,000 dedicated professionals at the FAA."
Meanwhile, the mounting pressure on Boeing is forcing the company to make tough decisions about its complex production schedule.
"We continue to work closely with the FAA and global regulators towards certification and the safe return to service of the MAX. We will continue to assess production decisions based on the timing and conditions of return to service, which will be based on regulatory approvals and may vary by jurisdiction," Boeing said in a statement.
-- Chris Isidore contributed to this report.