The 20-year-old joined the Clarets in July on a four-year deal after his contract at Old Trafford expired.
The tribunal committee are set to inform Burnley and United of its verdict in the next few days, and Clarets boss Brian Laws is confident of a positive outcome.
"We had a very good hearing and put our case across," Laws told the clubs official website.
"We remain hopeful of a positive outcome, but we must now await the verdict."
Eckersley, who progressed through the youth ranks at Old Trafford has made three appearances for Burnley this season.