The Cowboys met with Ryan on Friday and both sides came away impressed, according to a league source.
"Things went well," the source said.
There is mutual interest. and the sides are working toward a deal, but nothing will be finalized by Friday night. There is an expectation on both sides, however, that a deal will come together quickly.
If Ryan joins the Cowboys, it would leave the Arizona, Carolina, Denver and Cleveland (where Ryan is technically still employed) as the remaining teams still seeking a defensive coordinator. Dick Jauron is expected to be the Browns' next coordinator, and the Cardinals and Panthers continue to interview candidates.
Mangini was fired last week by the Browns after two underwhelming seasons. Ryan has one year remaining on his contract in Cleveland, but he isn't expected to be asked to return under new coach Pat Shurmur.