The Cowboys running back is thought to have contacted Eagles coach Chip Kelly on Wednesday and will meet the rest of his coaching staff before discussing a deal.
Murray enjoyed a hugely successful season with Dallas, putting up 1,894 yards and 13 TDs rushing for the team.
The arrival of Murray would represent a huge turnaround of fortunes for the Eagles after San Francisco 49ers running back Frank Gore turned his back on the team and decided to sign with the Indianapolis Colts instead.
Philadelphia turned their attention to Ryan Matthews, who signed from the San Diego Chargers on a three-year, $11.5m deal, although there are still questions about his injury-prone past.
New Eagles quarterback Sam Bradford ??? who traded for Nick Foles on Tuesday in one of the stranger deals of the off-season ??? said that he had tried to recruit Murray after playing alongside him at Oklahoma.
???I’ve been trying my hardest,??? he said. ???I’ve called him. I’ve texted him. I’ve done everything. We’ve been in communication. If we can somehow land him, it would be a great pickup.
“Not only is he a great player, he’s a great person. He’s a really close friend of mine, too. I think he can really help this ball club.”
The Cowboys are still believed to be considering a late offer for Murray, but Philly are front runners to land the 27-year-old.