The former Lyon man's unavailability, however will be keenly felt against the Red Devils, who possess the best defence in the Premier League.
United also have a number of injury complaints with holding midfielder Nemanja Matic a doubt.
Matic, who arrived at Old Trafford from Chelsea at the start of the season, was substituted after 55 minutes of United's 4-2 win at Watford in mid-week with Mourinho saying the Serbia global had a muscular problem.
"He is selected to go". "Let's see the feeling before the match, let's see the warm up, I hope he can (play)".
Jones has been out of action since limping off in the first half of England's match against Germany on November 10.
"I don't know if he can play", Mourinho admitted.
Southgate added: "We have got ongoing dialogue over this situation".
"Borg's coach was telling him that all the time: "One point at a time; think about just one point". Arsenal only really needed a point to seal United's fate, who were poor on the day and offered little threat to a team who had been unbeaten all season away from Highbury.
Raheem Sterling's goal six minutes into stoppage time earned a 2-1 victory over Southampton on Wednesday, three days after the same player had scored with seven minutes to spare to beat Huddersfield Town. "Yesterday I was watching an interesting movie, Borg v McEnroe".
Either he starts Alexis Sanchez at the tip of the Arsenal attack like he did in a game against Manchester City or he plays the Chilean in his usual role on the flank with Giroud leading the line as a traditional striker. It doesn't matter what comes after or what happens before. Arsene Wenger's side dominated the game, allowing just 30% of ball possession to the Terriers and completed 809 passes, even 477 more than Huddersfield. So now it is Arsenal. "Any other season 32 points after 14 games would put them in a fantastic position, so it is more about City being fantastic than United not doing well". "It is just Arsenal".
Mourinho was asked whether he provides the McEnroe to Wenger's Borg, but an amused United manager refused to make the comparison.