Garrett, the Cowboys' offensive coordinator, also has been courted by the Atlanta Falcons for their head coaching vacancy.
After interviewing with the Ravens and the Falcons the past two days, Garrett flew back to Dallas last night to contemplate his future.
If Garrett turns down the Ravens, they will turn their attention to Philadelphia Eagles secondary coach John Harbaugh, the source said. The Ravens contacted Harbaugh over the weekend to notify him that they wanted a second meeting, but the interview had not been scheduled as of last night.
Harbaugh, though, could be in Baltimore for an afternoon meeting because he is only a two-hour drive away.
Harbaugh, 45, has never been a head coach at any level but impressed the Ravens in his Jan. 8 visit.
The Ravens presumably want to name a coach by the end of the week.
Former Ravens defensive coordinator Rex Ryan and veteran head coach Marty Schottenheimer are no longer being considered by the Ravens, the source said.
If Garrett is no longer in the running in Atlanta, the Falcons could hire Ryan, who has been a favorite of owner Arthur Blank's.
Ravens officials tried to hire Garrett on Monday, visiting with him for seven hours. It is believed they talked to him about a four-year deal.
One league source said indications are Garrett could be leaning toward staying with Dallas.
Knowing his support from owner Jerry Jones, Garrett could eventually succeed head coach Wade Phillips. But giving Garrett a hefty raise could complicate matters in Dallas.
To keep Garrett with the Cowboys, Jones might have to pay Garrett as much as - or perhaps even more than - Phillips, who reportedly makes about $3 million a season.
According to The Dallas Morning News, Jones met with Garrett last night.
Ravens fans don't want Jason Garrett or John Harbaugh as the new head coach as much as they want someone else. Results of a poll at baltimoresun.com asking respondents to pick one of the two front-runners or some other candidate (1,782 responses as of last night):
Garrett .............24.4 percent
Harbaugh ......... 31.5 percent
Other................... 44.1 percent
For more coverage, go to baltimoresun.com/ravens.