Louis van Gaal revealed the duo, who missed Monday's FA Cup fifth round triumph over Preston North End, are available for selection at the Liberty Stadium.
Jones was forced off with a knee injury in the 3-1 win against Burnley on February 11, while Van Persie was nursing a knock.
The news leaves Michael Carrick (calf) as United's only absentee.
"They [Jones and van Persie] are fit to play," Van Gaal told MUTV. "Only Carrick is still injured.
"I think he shall train with the group next week. I expect that, but you never know. You'll have to wait and see."
Van Gaal will be wary of a Swansea side who spoiled his first competitive game as United boss, when Garry Monk's men won 2-1 at Old Trafford in the opening match of the season.
But the Dutchman pointed to United's "unbelievable" team spirit, which has helped them climb to third in the table on the back of a run of one defeat from their last 19 games in all competitions.
He added: "When you see the score of the last 19 or 20 matches, we are the best team in the league.
"I am never 100 per cent happy. We can play better than we are now, but the main thing is to win games.
"We've shown our team spirit many times this season. It's good for the fans and for me, as you see a player wants to follow you."
