Douglas made the comments when he met with reporters Wednesday for a news conference. Darnold has been linked to several teams this off-season as a potential trade candidate.
The Jets also have the No. 2 overall pick in the 2021 NFL Draft and could opt to select one of the top college quarterback prospects.
"I will answer the call if it's made," Douglas said of potential trade offers from other teams. "As I said, Sam, we think, is a dynamic player in this league with unbelievable talent who really has a chance to hit his outstanding potential. But if calls are made, I will answer them."
Douglas had a different perspective on Darnold in 2019, when he said the team would listen to trade offers for every player "unless it's a franchise quarterback."
Darnold, 23, completed 59.6% of his throws for 2,208 yards, nine touchdowns and 11 interceptions in 12 starts last season. The No. 3 overall pick in the 2018 NFL Draft has one year remaining on his rookie contract.
"Our timeline right now is just gathering as much information as we can as it pertains to free agency and this year's draft class," Douglas said. "We don't have a hard timeline in the immediate future to make a decision."
The 2021 NFL Draft is from April 29 to May 1. Clemson's Trevor Lawrence, BYU's Zach Wilson, Ohio State's Justin Fields, North Dakota State's Trey Lance, Alabama's Mac Jones and Florida's Kyle Trask are among the top quarterback prospects in the 2021 draft class.
The NFL free agency period starts March 17, the first day of the new league year.