Ian Rapoport of NFL Network noted it is typically "incredibly difficult" for teams to reach agreements by that date, but that is the Cowboys' stated goal.
The team could use the franchise tag on wide receiver Amari Cooper if it is able to sign Prescott to a long-term deal. Dallas is expected to use the transition tag on Cooper if it has to franchise Prescott.
Cowboys executive vice president Stephen Jones expressed optimism last week that a deal would get finalized.
"We want to get this done," Jones said, per Clarence E. Hill Jr. of the Star-Telegram. "Things are fixing to heat up. We want to put every foot forward and try to grind this out and get a deal done."
"I know he wants to get his contract in the rearview mirror, and we want it too," Jones continued. "We want him to be treated well, financially and respectfully. We are going to have a real urgency to get this done."
Prescott is coming off a career-best season that saw him set highs in passing yards (4,902) and touchdowns (30). The Cowboys finished with the No. 1 offense in football but finished 8-8 and missed the playoffs, leading to the departure of head coach Jason Garrett. Dallas hired former Green Bay Packers coach Mike McCarthy, who has heaped praise on Prescott since taking the job.
Prescott said he's removed himself from the situation and is allowing his agent to handle negotiations.
"It's in my agents' hands and the Cowboys' hands," Prescott told B/R's Mike Tanier. "I feel confident that something will be done. I'll continue to prepare for this season. No matter where I am or what the contract situation looks like, I'll be getting my body ready and preparing to play for the season."
If Prescott is franchised, the two sides will have until mid-July to hammer out a long-term contract. Prescott would have to play out the 2020 season on the franchise tag if they are unable to reach a deal by then—a scenario that neither side appears to want.