Todd, 56, was in third place Monday at the end of the second of the three phases that make up the individual eventing competition.
After two days of dressage, the competitors took part in a cross-country ride Monday and on Tuesday will go through the jumping competition.
Both individual and team medals will be at stake Monday, and if Todd can move ahead of Germany's Ingrid Klimke and Sweden's Sara Algotsson Ostholt, he will win his third gold medal in the event.
The first one came in Los Angeles in 1984 and he repeated as champion in Seoul four years later. Todd also won a bronze medal in Sydney 12 years ago.
Klimke and Algotsson Ostholt both had 39.30 points heading into the jumping while Todd stood at 39.50. Germany's Michael Jung was fourth at 40.60.
In the team portion of the event, Germany had the lead with 124.70 points, followed by Britain (130.20) and Sweden (131.40).
Monday's cross-country rides were marred by a fall suffered by Hawley Bennett-Awad of Canada.
The CBC reported Bennett-Awad was in stable condition at Royal London Hospital.
Workers accuse National Zoo of animal mismanagement
Video of Victoria’s Secret models trying to 'twerk' hits Instagram