The 26-year-old was held after appearing to throw three punches at Jones during the second half of Orient's 3-1 victory in the Sky Bet League One match at the County Ground on Saturday.
Wiltshire Police said the man, from Newbury in Berkshire, was arrested on Saturday on suspicion of pitch invasion and then further arrested on suspicion of affray.
Orient boss Russell Slade said that, although 24-year-old Jones was unhurt, he was angered by the incident, which Slade described as the worst he had seen since becoming a manager.
Swindon chairman Jed McCrory said the club would be launching a review of its safety measures.
Orient's chief executive Matt Porter wrote on Twitter: "Swindon confirm fan who ran on pitch and attacked Jamie was arrested and will be banned for life. We will also press for strongest penalty."
The FA will speak to officials from both Swindon and Orient before deciding whether to take any action.
A spokesman said: "We deplore scenes of this kind and would call for the authorities to take the appropriate actions if allegations of assault are proven.
"We will be speaking to both clubs involved to determine the facts surrounding the incident."
The spokesman also confirmed that the FA would push for a banning order from all grounds.
Wiltshire Police confirmed the arrested fan had been bailed.
A force spokesman said: "The 26-year-old male arrested at the Swindon Town match on November 23 has been released on police bail pending further inquiries.
"The man, from the Newbury area, was originally arrested on suspicion of pitch invasion and then further arrested on suspicion of affray.
"He was arrested at the County Ground during the Swindon Town game against Leyton Orient.
"It would be inappropriate to comment further at this stage."
- Crime & Justice
- Society & Culture
- pitch invasion
- Russell Slade