Cowboys owner and general manager Jerry Jones confirmed that he has had several meeting with Witten this week.
"He's not made any decisions that are definite at this time, we have no announcement today, as it pertains to Jason's future other than to say that he's a wonderful, valued member of our organization and our family".
Jones said the Cowboys would have no other comment on Witten's future the rest of the weekend, which includes the last two days of the National Football League draft Friday and Saturday.
Cowboys tight end Jason Witten recently said he plans to play until he's 40.
The Cowboys' all-time leader in receiving yards and receptions, who turns 36 next month, has seen a decrease in production in recent years. There's Geoff Swaim, who caught two passes for 25 yards last season; undrafted second-year player Blake Jarwin; and basketball player turned tight end (and almost defensive end) Rico Gathers.
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It didn't matter that James had to score almost half of Cleveland's points himself for this to be any sort of a competitive game. Luckily, James' efficiency and them going 26-of-27 from the free-throw stripe saved them in this one, along with their defense.
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Witten is also the franchise leader in consecutive games, starts and consecutive starts.
-The New England Patriots acquired tackle Trent Brown from the San Francisco 49ers, both teams announced. The Cowboys are still armed with nine remaining draft picks to choose. Day 2 of the NFL Draft?
Tartt was a second-round pick in 2015.
At No. 19 in the first round on Thursday night, the Cowboys had their pick of pass catchers.
A person with knowledge of the decision tells The Associated Press that tight end Antonio Gates will not return for a 16th season with the Chargers.
Background from SportsDay contributor Dane Brugler's draft guide: A four-star tight end recruit out of high school, Dalton Schultz helped lead Bingham to an undefeated 2013 season (14-0) and 5A state title with 31 catches for 512 yards and nine touchdowns as a senior - high school teammates with Utah defensive tackle Lowell Lotulelei.