Bechta tweeted that the move is likely to become official Thursday.
Chas Henry beat out McBriar, a two-time Pro Bowler in Dallas, for the Eagles' punting job. McBriar, who had offseason surgery to repair a nerve condition stemming from the drop foot that prematurely ended his 2011 season, averaged 41.3 yards on eight preseason punts.
"I suspect he will be snapped up quickly," Bechta tweeted.
The Cowboys opted not to re-sign McBriar, who averaged 45.3 yards per punt during his eight-season tenure in Dallas, after a late July workout because they didn't want to match the $300,000 in guaranteed money the Eagles were offering.
Chris Jones, McBriar's replacement, has averaged 44.7 yards on 13 punts this preseason.